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

Authors: Byung-Kwon Son; Do-Jin An; Joon-Ho Lee
Journal: Journal of Sensors
 (2020)

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## 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.


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## Body

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


## 2. LS-Based Location Estimate [10]

Letxi,yi and ϕi denote the ith sensor location and the AOA measurement at xi,yi. Given x=xi,yiT and ϕi for 1≤i≤N, where N is the number of sensor coordinates, we are to estimate xT=xT,yT, which denotes the true emitter location. x estimation of the emitter location can be written as [10]
(1)Ax=b,where
(2)A=sinϕ1−cosϕ1sinϕ2−cosϕ2⋮⋮sinϕN−cosϕNb=x1sinϕ1−y1cosϕ1x2sinϕ2−y2cosϕ2⋮xNsinϕN−yNcosϕN.The linear least-squared (LS) estimation algorithm for estimating the emitter location is briefly described. SinceA is not invertible, the following normal equation is obtained from (1) for the LS estimate:
(3)AHAx=AHb.The location estimate for the noiseless AOA measurements can be written as(4)x=AHA−1AHb.Noiseless AOA is denoted byϕi and noisy AOA is denoted ϕi+δϕi, where δϕi 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
(5)A′x′=b′,where A′ and b′ are defined as
(6)A′=sinϕ1+δϕ1−cosϕ1+δϕ1sinϕ2+δϕ2−cosϕ2+δϕ2⋮⋮sinϕN+δϕN−cosϕN+δϕN,(7)b′=x1sinϕ1+δϕ1−y1cosϕ1+δϕ1x2sinϕ2+δϕ2−y2cosϕ2+δϕ2⋮xNsinϕN+δϕN−yNcosϕN+δϕN.The normal equation of (7) is given by
(8)A′HA′x′=A′Hb′.The location estimate is given by(9)x′=A′HA′−1A′Hb′.


## 3. Approximation ofA′HA′ and A′Hb′

From (6) and (7), it is easy to show that the entries of A′HA′ and A′Hb′ can be expressed as
(10)A′HA′11=∑i=1Nsin2ϕi+δϕi,A′HA′12=A′HA′21=−12∑i=1Nsin2ϕi+δϕi,A′HA′22=∑i=1Ncos2ϕi+δϕi,A′Hb′1=∑i=1Nxisin2ϕi+δϕi−12yisin2ϕi+δϕi,A′Hb′2=−∑i=1N12xisin2ϕi+δϕi−yicos2ϕi+δϕi,where A′HA′ij and A′Hb′ij denote the entry at the ith row and the jth column of A′HA′ and A′Hb′, respectively. Note that A′HA′ is a 2×2 matrix and that A′Hb′ is a 2×1 matrix.LetA′HA′k and A′Hb′k denote the approximations of A′HA′ and A′Hb′ based on the kth-order Taylor series expansion, respectively:
(11)A′HA′≈A′HA′k=AHA+δAHAk,(12)A′Hb′≈A′Hb′k=AHb+δAHbk,where δAHAk and δAHbk are defined from
(13)δAHAk=AHAk−AHA,δAHbk=AHbk−AHb.If the first-orderK-approximation, corresponding to k=1, is applied, (11) and (12) can be written as
(14)A′HA′≈AHA+δAHAk=1,A′Hb′≈AHb+δAHbk=1.The explicit expressions ofδA′HA′k=1 and δA′Hb′k=1 based on the first-order Taylor series are derived in [16]
(15)δAHA11k=1=∑i=1Nδϕisin2ϕi,δAHA12k=1=−∑i=1Nδϕicos2ϕi,δAHA22k=1=−∑i=1Nδϕisin2ϕi,δAHb1k=1=∑i=1Nxiδϕisin2ϕi−yiδϕicos2ϕi,δAHb2k=1=∑i=1N−xiδϕicos2ϕi+yiδϕisin2ϕi.The explicit expressions ofδA′HA′k=2, δA′HA′k=3, δA′Hb′k=2, and δA′Hb′k=3 are derived in Appendices A and B, where k=2,3 denotes that the second- and third-order Taylor expansion has been adopted.


## 4. Error Bound for theK-Approximation

In this section, we describe the error bound due to various orders ofK-approximation for each of the entries of A′HA′k and A′Hb′k. In Table 1, all the terms of Taylor series expansion for each entries of A′HA′k and A′Hb′k 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 A′HA′11k.Table 1
Explicit expressions of each term of the Taylor series expansion.
ConstantFirst-order termSecond-order termThird-order termFourth-order termA′HA′11k∑i=1Nsin2ϕi∑i=1Nsin2ϕiδϕi∑i=1Ncos2ϕiδϕi2−23∑i=1Nsin2ϕiδϕi3−13∑i=1Ncos2ϕiδϕi4A′HA′12k−12∑i=1Nsin2ϕi−∑i=1Ncos2ϕiδϕi∑i=1Nsin2ϕiδϕi223∑i=1Ncos2ϕiδϕi313∑i=1Nsin2ϕiδϕi4A′HA′22k∑i=1Ncos2ϕi−∑i=1Nsin2ϕiδϕi−∑i=1Ncos2ϕiδϕi223∑i=1Nsin2ϕiδϕi313∑i=1Ncos2ϕiδϕi4A′Hb′1k∑i=1Nxisin2ϕi−yi2sin2ϕi∑i=1Nxisin2ϕi−yicos2ϕiδϕi∑i=1Nxicos2ϕi+yisin2ϕiδϕi2−23∑i=1N−xisin2ϕi+yicos2ϕiδϕi3−13∑i=1Nxicos2ϕi+yisin2ϕiδϕi4A′Hb′2k−∑i=1Nxi2sin2ϕi+yicos2ϕi−∑i=1Nxicos2ϕi−yisin2ϕiδϕi∑i=1Nxisin2ϕi+yicos2ϕiδϕi223∑i=1Nxicos2ϕi+yisin2ϕiδϕi3−13∑i=1Nxisin2ϕi+yicos2ϕiδϕi4Table 2
Taylor series and error bound forA′HA′11k.
Constant∑i=1Nsin2ϕiFirst-order term∑i=1Nsin2ϕiδϕiSecond-order term∑i=1Ncos2ϕiδϕi2Third-order term−23∑i=1Nsin2ϕiδϕi3Fourth-order term−13∑i=1Ncos2ϕiδϕi4Error boundFirst-order Taylor series∑i=1Nsin2ϕi+∑i=1Nsin2ϕiδϕi∑i=1Ncos2ϕiδϕi2Second-order Taylor series∑i=1Nsin2ϕi+∑i=1Nsin2ϕiδϕi+∑i=1Ncos2ϕiδϕi2−23∑i=1Nsin2ϕiδϕi3Third-order Taylor series∑i=1Nsin2ϕi+∑i=1Nsin2ϕiδϕi+∑i=1Ncos2ϕiδϕi2−23∑i=1Nsin2ϕiδϕi3−13∑i=1Ncos2ϕiδϕi4Note that the upper bound of absolute value of the error between the original function value and thenth-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, respectively.Similarly, Tables3–6 tabulate the same quantities for A′HA′12k=A′HA′21k, A′HA′22k, A′Hb′1k, and A′Hb′2k, respectively.Table 3
Taylor series and error bound forA′HA′12k.
Constant−12∑i=1Nsin2ϕiFirst-order-term−∑i=1Ncos2ϕiδϕiSecond-order-term∑i=1Nsin2ϕiδϕi2Third-order term23∑i=1Ncos2ϕiδϕi3Fourth-order term13∑i=1Nsin2ϕiδϕi4Error boundFirst-order Taylor series−12∑i=1Nsin2ϕi−∑i=1Ncos2ϕiδϕi∑i=1Nsin2ϕiδϕi2Second-order Taylor series−12∑i=1Nsin2ϕi−∑i=1Ncos2ϕiδϕi+∑i=1Nsin2ϕiδϕi223∑i=1Ncos2ϕiδϕi3Third-order Taylor series−12∑i=1Nsin2ϕi−∑i=1Ncos2ϕiδϕi+∑i=1Nsin2ϕiδϕi2+23∑i=1Ncos2ϕiδϕi313∑i=1Nsin2ϕiδϕi4Table 4
Taylor series and error bound forA′HA′22k.
Constant∑i=1Ncos2ϕiFirst-order term−∑i=1Nsin2ϕiδϕiSecond-order term−∑i=1Ncos2ϕiδϕi2Third-order term23∑i=1Nsin2ϕiδϕi3Fourth-order term13∑i=1Ncos2ϕiδϕi4Error boundFirst-order Taylor series∑i=1Ncos2ϕi−∑i=1Nsin2ϕiδϕi−∑i=1Ncos2ϕiδϕi2Second-order Taylor series∑i=1Ncos2ϕi−∑i=1Nsin2ϕiδϕi−∑i=1Ncos2ϕiδϕi223∑i=1Nsin2ϕiδϕi3Third-order Taylor series∑i=1Ncos2ϕi−∑i=1Nsin2ϕiδϕi−∑i=1Ncos2ϕiδϕi2+23∑i=1Nsin2ϕiδϕi313∑i=1Ncos2ϕiδϕi4Table 5
Taylor series and error bound forA′Hb′1k.
Constant∑i=1Nxisin2ϕi−yi2sin2ϕiFirst-order term∑i=1Nxisin2ϕi−yicos2ϕiδϕiSecond-order term∑i=1Nxicos2ϕi+yisin2ϕiδϕi2Third-order term−23∑i=1Nxisin2ϕi−yicos2ϕiδϕi3Fourth-order term−13∑i=1Nxicos2ϕi+yisin2ϕiδϕi4Error boundFirst-order Taylor series∑i=1Nxisin2ϕi−yi2sin2ϕi+∑i=1Nxisin2ϕi−yicos2ϕiδϕi∑i=1Nxicos2ϕi+yisin2ϕiδϕi2Second-order Taylor series∑i=1Nxisin2ϕi−yi2sin2ϕi+∑i=1Nxisin2ϕi−yicos2ϕiδϕi+∑i=1Nxicos2ϕi+yisin2ϕiδϕi2−23∑i=1Nxisin2ϕi−yicos2ϕiδϕi3Third-order Taylor series∑i=1Nxisin2ϕi−yi2sin2ϕi+∑i=1Nxisin2ϕi−yicos2ϕiδϕi+∑i=1Nxicos2ϕi+yisin2ϕiδϕi2+23∑i=1N−xisin2ϕi+yicos2ϕiδϕi3−13∑i=1Nxicos2ϕi+yisin2ϕiδϕi4Table 6
Taylor series and error bound forA′Hb′2k.
Constant−∑i=1Nxi2sin2ϕi+yicos2ϕiFirst-order term−∑i=1Nxicos2ϕi+yisin2ϕiδϕiSecond-order term∑i=1Nxisin2ϕi−yicos2ϕiδϕi2Third-order term23∑i=1Nxicos2ϕi+yisin2ϕiδϕi3Fourth-order term−13∑i=1Nxisin2ϕi−yicos2ϕiδϕi4Error boundFirst-order Taylor series−∑i=1Nxi2sin2ϕi−yicos2ϕi−∑i=1Nxicos2ϕi+yisin2ϕiδϕi∑i=1Nxisin2ϕi−yicos2ϕiδϕi2Second-order Taylor series−∑i=1Nxi2sin2ϕi−yicos2ϕi−∑i=1Nxicos2ϕi+yisin2ϕiδϕi+∑i=1Nxisin2ϕi−yicos2ϕiδϕi223∑i=1Nxicos2ϕi+yisin2ϕiδϕi3Third-order Taylor series−∑i=1Nxi2sin2ϕi−yicos2ϕi−∑i=1Nxicos2ϕi+yisin2ϕiδϕi+∑i=1Nxisin2ϕi−yicos2ϕiδϕi2+23∑i=1Nxicos2ϕi+yisin2ϕiδϕi3−13∑i=1Nxisin2ϕi−yicos2ϕiδϕi4


## 5. Approximation ofx′k

Substituting (11) and (12) in (7) results in
(16)AHA+δAHAkx′k=AHb+δAHbk,(17)x′k=AHA+δAHAk−1AHb+δAHbk.L-approximations of x′k=1 is denoted by x′k=1,l=1. In Appendix C, based on the perturbation of the solution of linear system, it is shown that x′k,l=1 can be written as
(18)x′k,l=1=xT+AHA−1δAHbk−δAHAkxT.


## 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 xT and yT denote the x-coordinate and y-coordinate of xT, respectively:
(19)xT=xTyT⋅TSimilarly,x′k,l=1 and y′k,l=1 denote the x-coordinate and y-coordinate of x′k,l=1, respectively:
(20)x′k=1,l=1=x′k,l=1y′k,l=1T.Euclidean distance betweenx′ and xT is given by
(21)dx′,xT=x′−xT2.Similarly, the distance betweenx′k,l=1 and xT is written as
(22)dx′k,l=1,xT=x′k,l=1−xT2.Substituting (18) in (22) yields
(23)x′k,l=1−xT2=AHA−1δAHbk−δAHAkxT2.From (23), the MSE of the location estimate is
(24)Ex′k,l=1−xT22=EδAHbk−δAHAkxTHAHA−1HAHA−1δAHbk−δAHAkxT.LetD and H be defined as
(25)D≡δAHbk−δAHAkxT=δAHb1k−δAHA11kxT+δAHA12kyTδAHb2k−δAHA21kxT+δAHA22kyT,(26)H≡AHA−1TAHA−1=AHA11−1AHA12−1AHA21−1AHA22−12.In terms ofD in (25) and H in (26), the MSE in (24) can be rewritten as
(27)Ex′k,l=1−xT22=EDTHD=H11ED12+H12+H21ED1D2+H22ED22=∑p=12∑q=12HpqEDpDq.where Di denote and the ith entry of D and Hij denote the entry at the ith row and thejth column of H, respectively.ED12, ED1D2, and ED22 are explicitly expressed as
(28)ED12=EδAHb1k2+xTEδAHA11k2+yT2EδAHA12k2+2xTyTEδAHA11kδAHA12k−2xTEδAHb1kδAHA11k−2yTEδAHb1kδAHA12k,(29)ED1D2=EδAHb1kδAHb2k−xTEδAHb1kδAHA21k−yTEδAHb1kδAHA22k−xTEδAHb2kδAHA11k−yTEδAHb2kδAHA12k+xT2EδAHA11kδAHA21k+xTyTEδAHA11kδAHA22k+xTyTEδAHA12kδAHA21k+yT2EδAHA12kδAHA22k,(30)ED22=EδAHb2k2+xTEδAHA21k2+yT2EδAHA22k2+2xTyTEδAHA21kδAHA22k−2xTEδAHb2kδAHA21k−2yTEδAHb2kδAHA22k,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 Table7.Table 7
Summary of the performance analysis of the AOA-based localization algorithm using the LS approach.
DataEstimate of emitter locationAHA,AHbxT=AHA−1AHbA′HA′,A′Hb′x′=A′HA′−1A′Hb′A′HA′k=AHA+δAHAkA′Hb′k=AHb+δAHbkx′k=A′HA′k−1A′Hb′k=AHA+δAHAk−1AHbk+δAHbkxTx′k,l=1=xT+δxk,l=1=xT+AHA−1δAHbk−δAHAkxT


## 7. Results and Discussion

Trajectory of sensor locations is given in Figure1.Figure 1
Geometry of the localization.
(a)
Linear trajectory(b)
Circular trajectorySimulation parameters are as follows:Speed of sensor: 0.280 km/secSampling interval: 2 secNumber of sensor locations: 100Number of Monte Carlo simulation:S=1000The empirical MSE ofx′, x′k=1 and x′k=1,l=1 is defined as
(31)SimulationEx′−xT2=1S∑s=1Sxs′−xT2+ys′−yT2,(32)SimulationEx′k=1−xT2=1S∑s=1Sx′sk=1−xT2+y′sk=1−yT2,(33)SimulationEx′k=1,l=1−xT2=1S∑s=1Sx′sk=1,l=1xT2+y′sk=1,l=1−yT2,where the lower script s denotes the estimate associated with the sth repetition out of S repetitions.Similarly, the empirical MSEs ofx′k=2, x′k=2,l=1, x′k=3, and x′k=3,l=1 are given by
(34)SimulationEx′k=2−xT2=1S∑s=1Sx′sk=2−xT2+y′sk=2−yT2,(35)SimulationEx′k=2,l=1−xT2=1S∑s=1Sx′sk=2,,l=1−xT2+y′sk=2,,l=1−yT2,(36)SimulationEx′k=3−xT2=1S∑s=1Sx′sk=3−xT2+y′sk=3−yT2,(37)SimulationEx′k=3,l=1−xT2=1S∑s=1Sx′sk=3,,l=1−xT2+y′sk=3,,l=1−yT2.x′ in (31) is defined in (9). x′k=1 and x′k=1,l=1 in (32) and (33) are given by (17) and (18) with k=1. Similarly, x′k=2 and x′k=2,l=1 in (34) and (35) are given by (17) and (18) with k=2. x′k=3 and x′k=3,l=1 in (36) and (37) are given by (17) and (18) with k=3.The results withx′k=1,l=1 can be calculated using (9) and (18), respectively. In Figures 2(a)–2(c), we illustrate how the location estimates are distributed with k=1, k=2, and k=3 for the linear trajectory. It is clear that x′k=2,l=1 is more accurate than x′k=1,l=1 and that x′k=3,l=1 is more accurate than x′k=2,l=1. Similarly, x′k=2,l=1 is closer to x′ than x′k=1,l=1 and x′k=3,l=1 is closer to x′ than x′k=2,l=1. Similar observations can be made in Figure 3, where the results for the circular trajectory have been illustrated.Figure 2
Distribution of estimates for the linear trajectory: (a) first-orderK-approximation, (b) second-order K-approximation, and (c) third-order K-approximation.
(a)(b)(c)Figure 3
Distribution of estimates for the circular trajectory: (a) first-orderK-approximation, (b) second-order K-approximation, and (c) third-order K-approximation.
(a)(b)(c)Linear trajectory in Figure1(a) is considered to get actual error and error bound in Figure 4. In Figure 4(a), the actual errors for A′HA′11k 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(b) represents corresponding results for A′Hb′1k, respectively.Figure 4
Error bound: (a)A′HA′11k; (b) A′Hb′1k.
(a)(b)The results validating the derived MSEs are illustrated in Figures5 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.Figure 5
MSE of target location estimate for the linear trajectory: (a) first-orderK-approximation, (b) second-order K-approximation, and (c) third-order K-approximation.
(a)(b)(c)Figure 6
MSE of target location estimate for the circular trajectory: (a) first-orderK-approximation, (b) second-order K-approximation, and (c) third-order K-approximation.
(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 Figures5(a) and 5(b) for the linear trajectory. The results in Figure 5(a) with ′SimulationEx′−x2′, ′SimulationEx′k=1−x2′, ′SimulationEx′k=1,l=1−x2′, and ′AnalyticEx′k=1,l=1−x2′ are obtained from (31), (32), (33) and (27), respectively.′SimulationEx′−x2′ is not equal to ′SimulationEx′k=1−x2′, since K-approximation is used to get ′SimulationEx′k=1−x2′ from ′SimulationEx′−x2′. Similarly, ′SimulationEx′k=1−x2′ is not equal to ′SimulationEx′k=1,l=1−x2′, since L-approximation is adopted to get ′SimulationEx′k=1,l=1−x2′ from ′SimulationEx′k=1−x2′. Although, they are not exactly equal, it is clear from Figure 5 that ′SimulationEx′−x2′ is approximately equal to ′SimulationEx′k=1−x2′ and that ′SimulationEx′k=1−x2′ is approximately equal to ′SimulationEx′k=1,l=1−x2′.It should be noted that′SimulationEx′k=1,l=1−x2′ and ′AnalyticEx′k=1,l=1−x2′ show excellent agreements, which validates (18) with k=1. Therefore, we can use analytically derived expression ′AnalyticEx′k=1,l=1−x2′ to see how ′SimulationEx′−x2′.The same observations can be made in Figure5(b). The results in Figure 5(b) with ′SimulationEx′−x2′, ′SimulationEx′k=2−x2′, ′SimulationEx′k=2,l=1−x2′, and ′AnalyticEx′k=2,l=1−x2′ are obtained from (31), (34), (35), and (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 K-approximation. The results in Figure 5(b) with ′SimulationEx′−x2′, ′SimulationEx′k=3−x2′, ′SimulationEx′k=3,l=1−x2′, and ′AnalyticEx′k=3,l=1−x2′ are obtained from (31), (36), (37), and (27), respectively.To reduce the error due toK-approximation, the second-order Taylor series can be used, which is illustrated in Figure 5(b). In Figure 5(a), difference between ′SimulationEx′−x2′ and ′SimulationEx′k=1−x2′ is much greater than that between ′SimulationEx′k=1−x2′ and ′SimulationEx′k=1,l=1−x2′ for large standard deviation of AOA estimation. On the other hand, difference between ′SimulationEx′−x2′ and ′SimulationEx′k=1−x2′ is much smaller than that between ′SimulationEx′k=1−x2′ and ′SimulationEx′k=1,l=1−x2′ for large standard deviation of AOA estimation in Figure 5(c). Actually, ′SimulationEx′−x2′ and ′SimulationEx′k=3−x2′ in Figure 5(c) is approximately equal for all standard deviation of AOA estimation error.In Figure6(a), it is clear that the difference between ′SimulationEx′−x2′ and ′SimulationEx′k=1−x2′ gets larger with the increase of standard deviation of AOA estimation, which implies that the error due to the first-order K-approximation becomes greater with the increase of standard deviation of AOA. ′SimulationEx′−x2′ and ′SimulationEx′k=1−x2′ are approximately equal for standard deviation of σϕ=0.2154∘, which implies that error due to K-approximation is very small for standard deviation of σϕ=0.2154∘. The difference between ′SimulationEx′k=1−x2′ and ′SimulationEx′k=1,l=1−x2′ quantifies how much error occurs due to L-approximation. The difference between ′SimulationEx′k=1−x2′ and ′SimulationEx′k=1,l=1−x2′ is negligible for standard deviation of σϕ=0.2154∘, which means that L-approximation results in approximately no error for standard deviation of σϕ=0.2154∘. Comparing ′SimulationEx′−x2′, ′SimulationEx′k=1−x2′, and ′SimulationEx′k=1,l=1−x2′, the first-order K-approximation induces much larger error than L-approximation.The error due toK-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 ′SimulationEx′−x2′ and ′SimulationEx′k=2−x2′ for the second-order K-approximation is smaller than that between ′SimulationEx′−x2′ and ′SimulationEx′k=1−x2′ for the first-order K-approximation.The error due toK-approximation can be made smaller by adopting the second-order K-approximation, which is illustrated in Figure 6(c). In Figure 6(c), it is shown that ′SimulationEx′−x2′ and ′SimulationEx′k=3−x2′ show agreement for all standard deviation of AOA estimation, which is quite different from ′SimulationEx′−x2′ and ′SimulationEx′k=1−x2′ 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 toK-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.



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*Source: 9346142-2020-08-18.xml*