Performance Analysis of AOA-Based Localization Using the LS Approach: Explicit Expression of Mean-Squared Error

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.

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 K-approximation and L-approximation have been employed to derive the mean-squared error (MSE) of location estimate.
Taylor series expansion has been used in K-approximation. K-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].

LS-Based Location Estimate [10]
Let ðx i , y i Þ and ϕ i denote the ith sensor location and the AOA measurement at ðx i , y i Þ. Given x = ðx i , y i Þ T and ϕ i for 1 ≤ i ≤ N, where N is the number of sensor coordinates, we are to estimate x T = ðx T , y T Þ, 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 A 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 ϕ i and noisy AOA is denoted ϕ i + δϕ i , where δϕ i denotes an error in AOA measurement. Under these conditions, x′ denotes an estimate of emitter location. The location estimate can be obtained from the least squares solution of where A ′ and b ′ are defined as The explicit expressions of δðA ′ H A ′ Þ ðk=1Þ and δ based on the first-order Taylor series are derived in [16] The explicit expressions of δðA′ H A′Þ , δðA′ H A′Þ are derived in Appendices A and B, where k = 2, 3 denotes that the second-and thirdorder Taylor expansion has been adopted.

Error Bound for the K-Approximation
In this section, we describe the error bound due to various orders of K-approximation for each of the entries of . Note that the upper bound of absolute value of the error between the original function value and the nth-order Taylor series is given by the n + 1th-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

Approximation of x ′
Substituting (11) and (12) in (7) results in L-approximations of x ′ ðk=1Þ is denoted by x ′ ðk=1,l=1Þ . In Appendix C, based on the perturbation of the solution of linear system, it is shown that x ′ ðk,l=1Þ can be written as 3 Journal of Sensors

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 x T and y T denote the x-coordinate and y-coordi-nate of x T , respectively: Similarly, x ′ ðk,l=1Þ and y ′ ðk,l=1Þ denote the x-coordinate

Error bound
First-order Taylor series Journal of Sensors and y-coordinate of x ′ ðk,l=1Þ , respectively: Euclidean distance between x′ and x T is given by Similarly, the distance between x ′ ðk,l=1Þ and x T is written as ð22Þ From (23), the MSE of the location estimate is

Error bound
First-order Taylor series Second-order Taylor series Third-order Taylor series Journal of Sensors

Error bound
First-order Taylor series 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 thirdorder Taylor series have been employed in Appendices D, E, and F, respectively. The summary of the localization algorithm is tabulated in Table 7.

Results and Discussion
Trajectory of sensor locations is given in Figure 1.
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: S = 1000 The empirical MSE of x ′ , x ′ ðk=1Þ and x′ ðk=1,l=1Þ is defined as Simulation Simulation E x′ Table 7: Summary of the performance analysis of the AOA-based localization algorithm using the LS approach.
The results with x ′ ðk=1,l=1Þ can be calculated using (9)    and x ′ ðk=3,l=1Þ is closer to x ′ than x ′ ðk=2,l=1Þ . Similar observations can be made in Figure 3, where the results for the circular trajectory have been illustrated. Linear trajectory in Figure 1(a) is considered to get actual error and error bound in Figure 4. In Figure 4(a), the actual 11 and their error bounds associated with the first-order Taylor series for 50 repetitions are illustrated. Note that the x-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 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 The same observations can be made in Figure 5(b). The results in Figure 5 (27), respectively. Note that the results in Figure 5(b) are for the second-order K-approximation and that the results in Figure 5(c) are for the third-order Kapproximation. The results in Figure 5(b) with ′ Simulation     Journal of Sensors To reduce the error due to K-approximation, the secondorder Taylor series can be used, which is illustrated in Figure 5(b). In Figure 5( Þ ′ , the first-order K-approximation induces much larger error than L-approximation. The error due to K-approximation can be reduced by adopting the second-order Taylor series in the K-approximation, which is illustrated in Figure 6(b). Compared with the results in Figure 6(b), the difference between ′ Simulation Eðkx ′ − xk 2 Þ ′ and ′ Simulation Eðkx ′ ðk=2Þ − xk 2 Þ ′ for the second-order K-approximation is smaller than that between ′Simulation Eðkx′ − xk 2 Þ′ and ′Simulation E ðkx′ ðk=1Þ − xk 2 Þ′ for the first-order K-approximation.
The error due to K-approximation can be made smaller by adopting the second-order K-approximation, which is illustrated in Figure 6(c). In Figure 6( Þ ′ in Figure 6(a).

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 meansquared 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 K -approximation and L-approximation is highly dependent on the noise variance, the scheme is more useful for high SNR. To reduce the error due to K-approximation and L -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 A ′ H A ′ and
The second-order Taylor expansion-based approximation of the entries of A ′ H A ′ and A ′ H b ′ is presented. ðA:1Þ

B. Third-Order Approximation of A ′ H A ′ and
The third-order Taylor expansion-based approximation of the entries of A ′ 2Þ denote an unperturbed linear system with C ∈ ℝ n×n , G ∈ ℝ n and x ∈ ℝ n . Consider a perturbed linear system, where F ∈ ℝ n×n and f ∈ ℝ n . If C is nonsingular, then it is clear that xðεÞ is differentiable in a neighborhood of zero. Let the first derivative of xðεÞ be denoted by _ xðεÞ. Differentiation of (C.2) yields x ≡ x ε = 0 ð Þ: ðC:5Þ Using (C.5) in (C.4) results in Solving for _ xðε = 0Þ yields First-order approximation of xðεÞ based on the Taylor expansion can be written as Comparing (C.2) and (16), εF and εf in (C.2) correspond to δðA H AÞ ðk=1Þ 11 and δðA H bÞ ðk=1Þ 11 in (16) sin 2 2ϕ i :

ðD:2Þ
Since the other entries can be derived similarly, we list the final expressions for the other entries: 17 Journal of Sensors 18 Journal of Sensors ðF:4Þ ðF:5Þ

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.