On the efficiency of a bearings-only instrumental variable estimator for target motion analysis

ARTICLE IN PRESS

Signal Processing 85 (2005) 481–490 www.elsevier.com/locate/sigpro

On the efficiency of a bearings-only instrumental variable estimator for target motion analysis Kutluyıl Dog˘anc- ay School of Electrical and Information Engineering,University of South Australia, Mawson Lakes, SA 5095, Australia Received 1 March 2004; received in revised form 14 October 2004

Abstract The maximum-likelihood (ML) estimator for bearings-only target motion analysis does not admit a closed-from solution and must be implemented iteratively. Iterative ML estimators require an initialization close to the true solution to avoid divergence. Recently a closed-form asymptotically unbiased instrumental variable estimator has been proposed to alleviate the convergence problems associated with iterative ML estimators. This paper establishes the asymptotic efficiency of the closed-form instrumental variable estimator by showing that its error covariance matrix approaches the Cramer–Rao lower bound for sufficiently small bearing noise as the number of measurements tends to infinity. r 2004 Elsevier B.V. All rights reserved. Keywords: Target motion analysis; Bearings-only target tracking; Asymptotic efficiency; Maximum likelihood; Pseudolinear estimator; Bias compensation; Instrumental variables

1. Introduction The objective of bearings-only target motion analysis is to estimate the position, velocity and acceleration of a target from noisy bearing angle measurements collected by a moving observer. The moving observer, which is also known as the ownship, can be an aircraft, a ship or an unmanned aerial vehicle (UAV). Target motion analysis or target tracking is an important Tel.: +61 8 8302 3984; fax: +61 8 8302 3384.

E-mail address: [email protected] (K. Dog˘anc- ay).

practical problem with applications in radar, sonar and mobile communications, to name but a few. Bearings-only target tracking has been an active area of research for several decades. Given the nonlinear relationship between the bearing measurements and the target location, the target tracking problem can be solved by the extended Kalman filter (EKF) [1]. The EKF algorithm is known to exhibit divergence problems in Cartesian coordinates [1], and a remedy has been found by implementing the EKF in modified polar coordinates [2]. Because of the recursive nature of the EKF algorithms, good initialization is a must in order to avoid divergence [14]. The

0165-1684/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2004.10.014

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Nomenclature a b dk d^k F Fo G JðiÞ K Mk nk mk N g

constant target acceleration ‘‘observation’’ vector of the pseudolinear estimator target range at time tk bias-compensated pseudolinear estimate of d k ‘‘data’’ matrix of the pseudolinear estimator noise-free F instrumental variable matrix Jacobian matrix of estimated bearing error diagonal covariance matrix of bearing noise matrix that transforms target motion parameters to target location at time tk bearing noise unit vector orthogonal to true bearing line at time tk the number of bearing measurements collected over the period ½0; TÞ noise vector obtained from nonlinear

maximum-likelihood (ML) estimator for target tracking [24,20] does not admit a closed-form solution and is therefore implemented as an iterative search algorithm. Iterative ML estimators are not only computationally expensive, but they also exhibit convergence difficulties unless initialized close to the solution. A least-squares (LS) estimator with closed-form solution, which is referred to as the pseudolinear estimator, was developed in [15]. Despite its simplicity and low computational complexity, the pseudolinear estimator suffers from severe bias [3,20]. To overcome the bias of the pseudolinear estimator, two iterative approaches have been developed, viz., the modified instrumental variable (MIV) estimation algorithm [20] and the recursive instrumental variable (IV) estimator [4]. Neither of these IV algorithms have a closed-form solution since they rely on an iterative process to compute an instrumental variable matrix. They also require appropriate initialization in order to avoid diver-

p0 pk rk s2n tk T yk y~ k uk v0 W Wo n n^ BCLS n^ LS n^ MIV n^ ML n^ WIV

functions of the bearing noise initial target location target location vector at time tk observer location vector at time tk bearing noise variance bearing measurement time instant bearing observation time interval bearing angle at time tk bearing measurement at time tk unit vector orthogonal to measured bearing line at time tk initial target velocity estimated weighting matrix for n^ WIV true weighting matrix for n^ WIV target motion parameter vector bias-compensated pseudolinear estimate of n pseudolinear estimate of n iterative modified instrumental variable estimate of n maximum-likelihood estimate of n closed-form weighted instrumental variable estimate of n

gence. The convergence properties of iterative bearings-only tracking algorithms are analyzed in detail in [13]. Several unbiased closed-form solutions have been proposed. Some of the key algorithms in this category are the reduced-bias closed-form tracker [11], the efficient unbiased estimator [23], and the closed-form reduced-bias pseudolinear estimator [19]. The first algorithm requires large range-tobaseline ratio and small bearing noise for unbiasedness, and the last two algorithms assume a multi-leg constant-velocity trajectory for the observer. Recently, we have proposed a new asymptotically unbiased closed-form tracking algorithm based on instrumental variables and a biascompensated pseudolinear estimator [6]. This algorithm considers a more general target motion model than the previous algorithms, which permits the target to have a constant acceleration. It also makes no restrictive assumptions about the

ARTICLE IN PRESS K. Dog˘anc- ay / Signal Processing 85 (2005) 481–490

observer trajectory or the range-to-baseline ratio. A version of the closed-form IV estimator considered in this paper has been successfully applied to stationary target localization in a previous work [5]. The main contribution of this paper is to establish the asymptotic efficiency of the closed-form IV tracking algorithm under the assumption of small bearing noise. The paper is organized as follows. Section 2 describes the target tracking problem. An overview of the key LS target tracking algorithms as well as the closed-form IV estimator is provided in Section 3. Asymptotic efficiency of the closed-form IV estimator is established in Section 4. Section 5 presents numerical simulation results to corroborate the findings of the paper. Conclusions are drawn in Section 6.

f0; 1; . . . ; N  1g; using the nonlinear equation yk ¼ tan1



In the two-dimensional bearings-only target tracking problem, an observer collects bearing angles of the target yk at observer locations rk as it moves along an observer trajectory as depicted in Fig. 1. The objective of bearings-only target tracking is to estimate the target location pk from N noisy bearing measurements taken at observer locations rk at discrete time instants k 2

θk rk Observer

0

x

Fig. 1. Two-dimensional bearings-only target tracking geometry.

t2k a 2

ð2aÞ ð2bÞ

where p0 and v0 are the initial target location and the initial velocity at k ¼ 0; respectively, a is the constant target acceleration, and 2 3 " # p0 1 0 tk 0 12 t2k 0 6 7 Mk ¼ v0 5: ; n ¼ 4 0 1 0 tk 0 12 t2k a

Observer Trajectory

0

(1)

The target moves at a constant acceleration, which subsumes the cases of constant-velocity and stationary targets. The yk are measured at regular time instants tk ¼ kT=N; k ¼ 0; . . . ; N  1; where T is the observation time interval. Increasing N therefore results in finer time separation between successive bearing measurements over the time interval ½0; TÞ: The target location at time tk is given by the kinematic equation [12]

¼ M k n;



pk

k ¼ 0; . . . ; N  1;

pk ¼ p0 þ t k v 0 þ

Target Trajectory Target

Dyk ; Dxk

where Dyk ¼ py;k  ry;k and Dxk ¼ px;k  rx;k with pk ¼ ½px;k ; py;k T and rk ¼ ½rx;k ; ry;k T denoting the target location vector and the observer location vector at discrete time instant k, respectively. The following assumptions are made about the target tracking problem:

2. Bearings-only target tracking problem and assumptions

y

483



Here n is the 6 1 target motion parameter vector to be estimated. The bearing measurements are corrupted by i.i.d. zero-mean Gaussian noise: y~ k ¼ yk þ nk ; (3) where the y~ k ; k ¼ 0; . . . ; N  1; are the bearing measurements and nk is a Gaussian random variable with zero mean and variance s2n : The bearing noise variance is assumed to be known a priori for the bias-compensated pseudolinear estimator in Section 3.2. The target is assumed to be observable so that the target motion vector n can be estimated uniquely from the available bearing

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measurements. This requires the observer to maneuver while collecting bearing measurements [18,22,7].

3. Overview of LS estimators 3.1. ML estimator The maximum likelihood estimate of the target motion parameters n^ ML is obtained from maximization of the log-likelihood function for the bearing measurements, which can be written as [6] n^ ML ¼ arg min J ML ðnÞ;

(4)

n2R6

where J ML ðnÞ is the ML cost function J ML ðnÞ ¼ eT ðnÞK 1 eðnÞ:

(5)

Here K ¼ s2n I is the N N diagonal covariance matrix of the bearing measurement noise and eðnÞ is the N 1 error vector 2 6 6 eðnÞ ¼ 6 6 4

y~ 0  ffðM 0 n  r0 Þ y~ 1  ffðM 1 n  r1 Þ .. .

3 7 7 7; 7 5

y~ N1  ffðM N1 n  rN1 Þ where ff denotes the angle of its vector argument (i.e., the bearing angle). The ML estimator is asymptotically unbiased and efficient (i.e., it obtains the Cramer–Rao lower bound as N ! 1). The minimization of J ML ðnÞ over n is a nonlinear LS problem without a closed-form solution. Gradient or downhill simplex-based iterative search algorithms such as the steepest descent algorithm [10], the Gauss-Newton (GN) algorithm [16] and the Nelder–Mead simplex algorithm [21] can be used to obtain a numerical ML solution. The gradient-based search algorithms aim to solve @J ML ðnÞ ¼ 0: (6) @n n¼n^ ML

The ML cost function has been shown to have a unique global minimum with no local minima [13]. This implies that the gradient-based search algorithms are globally convergent subject to an appropriate selection of the stepsize. Despite being globally convergent, these algorithms usually have slow convergence rates. On the other hand, the GN algorithm given below offers almost quadratic convergence in the vicinity of its solution ^ þ 1Þ ¼ nðiÞ ^  ðJ T ðiÞK 1 JðiÞÞ1 J T ðiÞ nði ^ i ¼ 0; 1; . . . ; K 1 eðnðiÞÞ;

ð7Þ

where JðiÞ is the Jacobian of eðnÞ evaluated at ^ n ¼ nðiÞ: 3 2 ½sin f0 ðiÞ;  cos f0 ðiÞ M 0 7 6 ^  r 0 k2 kM 0 nðiÞ 7 6 7 6 7 6 ½sin f1 ðiÞ;  cos f1 ðiÞ M 1 7 6 7 6 ^  r 1 k2 kM 1 nðiÞ 7; JðiÞ ¼ 6 7 6 7 6 . 7 6 . . 7 6 7 6 4 ½sin fN1 ðiÞ;  cos fN1 ðiÞ M N1 5 ^  rN1 k2 kM N1 nðiÞ ^  rk Þ: fk ðiÞ ¼ ffðM k nðiÞ Here k  k2 denotes the Euclidean norm. The GN algorithm requires the use of an initial guess that is sufficiently close to the solution. If it is initialized far from the solution, it may exhibit divergence. Since each GN iteration involves matrix inversion, the computational complexity of the GN algorithm can become quite demanding if a large number of GN iterations is required.

3.2. Pseudolinear estimator and bias compensation Rewriting (3) as sin ðy~ k  nk Þ Dyk ¼ cos ðy~ k  nk Þ Dxk and lumping the noise terms together leads to the matrix equation Fn ¼ b þ g;

(8)

ARTICLE IN PRESS K. Dog˘anc- ay / Signal Processing 85 (2005) 481–490

where 2

uT0 M 0

6 6 6 F¼6 6 6 4 2

uT1 M 1 .. .

3

2

7 7 7 7 7 7 5

6 6 6 b¼6 6 6 4

uTN1 M N1 Z0

3

;

uT0 r0 uT1 r1 .. . uTN1 rN1

N 6

7 6 6 Z1 7 7 6 7 g¼6 : 6 .. 7 6 . 7 5 4 ZN1 N 1

developed in [20] as an approximation to the ML estimator. The iterative MIV estimator is obtained from the normal equations

3 7 7 7 7 7 7 5

;

N 1

where 2

" u^ k ðiÞ ¼

3

u^ T0 ðiÞM 0

6 6 6 GðiÞ ¼ 6 6 6 4

ð10aÞ

n2R6

7 7 7 7; 7 7 5

u^ T1 ðiÞM 1 .. . u^ TN1 ðiÞM N1 sin ffðM k n^

MIV ðiÞ

 rk Þ

# ð14Þ

 cos ffðM k n^ MIV ðiÞ  rk Þ

and

¼ ðF T FÞ1 F T b;

ð10bÞ

which is referred to as the pseudolinear estimator [20]. The matrix F T F is assumed to be nonsingular to make the target observable. The pseudolinear estimator is biased even for N ! 1: A bias-compensated pseudolinear estimator was developed in [6] by subtracting an instantaneous estimate of the bias from the pseudolinear estimate: n^ BCLS ¼ n^ LS þ m2 ðF T FÞ1

(12)

n^ MIV ði þ 1Þ ¼ ðG T ðiÞW 1 ðiÞFÞ1 G T ðiÞW 1 ðiÞb; i ¼ 0; 1; . . . ; ð13Þ

Here uk ¼ ½sin y~ k ;  cos y~ k T is the orthogonal unit bearing vector and Zk ¼ d k sin nk is a nonlinear transformation of the bearing noise multiplied by the target range d k ¼ kpk  rk k2 : An LS solution to the approximate matrix equation FnEb is given by

N 1 X

M Tk

k¼0

ðM k n^ LS  rk Þ; 2

F T F n^ LS ¼ F T b by replacing F T with G T ðiÞW 1 ðiÞ:

ð9Þ

n^ LS ¼ arg min kFn  bk22

485

ð11Þ

2

where m ¼ Efsin nk g: For small bearing noise, the bias-compensated pseudolinear estimator was shown to exhibit a significantly reduced bias compared to the pseudolinear estimator [6].

2 6 6 6 6 WðiÞ ¼ 6 6 6 4

2 d^0 ðiÞ

0 d^k ðiÞ ¼ kM k n^

0 2 d^ 1 ðiÞ

..

. 2 d^ N1 ðiÞ

MIV ðiÞ

3 7 7 7 7 7; 7 7 5

 rk k 2 :

ð15Þ

The MIV estimator given by (13) is an extension of the original MIV estimator to constant-acceleration target tracking. A major disadvantage of the MIV estimator is its high computational complexity which grows linearly with the number of iterations. 3.4. Asymptotically unbiased WIV estimator

3.3. Iterative MIV estimator

The closed-form weighted IV (WIV) estimator developed in [6] is given by

In order to remove the bias of the pseudolinear estimator, an iterative MIV estimator was

n^ WIV ¼ ðG T W 1 FÞ1 G T W 1 b;

(16)

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where G is the instrumental variable matrix 3 2 u^ T0 M 0 7 6 6 u^ T1 M 1 7 7 6 (17) G ¼6 7 .. 7 6 . 5 4 u^ TN1 M N1 and W is the weighting matrix 3 2 2 d^0 0 7 6 7 6 ^2 7 6 d 1 7: 6 W ¼6 7 .. 7 6 . 5 4 2 0 d^

1 s2n ðF To W 1 o F oÞ ;

(22)

where F o is the noise-free version of F 2 3 mT0 M 0 6 mT M 7  1 6 7 1 sin yk 6 7 Fo ¼ 6 .. 7; mk ¼  cos y k 4 5 .

(23)

m TN1 M N1

(18)

N1

In above expressions, u^ k is the orthogonal unit bearing vector estimate " # sin y^ k u^ k ¼ ; k ¼ 0; . . . ; N  1 (19)  cos y^ k with y^ k ¼ ffðM k n^ BCLS  rk Þ; and d^k is the range estimate d^ k ¼ kM k n^ BCLS  rk k2 ;

CRLB is given by [20]

k ¼ 0; . . . ; N  1:

(20)

Both u^ k and d^k are obtained from the bias compensated pseudolinear estimator in (11). It was shown in [6] that the WIV estimator is asymptotically unbiased, i.e., Efn^ WIV g ¼ n as N ! 1: In Section 4, we examine the asymptotic error covariance matrix of the WIV estimator.

and W o is the weighting matrix obtained from the true range values 3 2 2 0 d0 7 6 7 6 d 21 7 6 (24) Wo ¼ 6 7: .. 7 6 . 5 4 0 d 2N1 Assuming that G T W 1 F=N is nonsingular, i.e., the target is observable, the probability limit of C WIV in (21) can be written as

1 1 G T W 1 F C WIV ¼ plim N N G T W 1 ggT W 1 G N

T 1 1 ! F W G ; N



ð25Þ

where plim denotes the probability limit and is defined by [17] ^ ^  x j4g ¼ 0 plim xðNÞ ¼ x () lim PfjxðNÞ N!1

4. Asymptotic efficiency of the WIV estimator For finite N, the error covariance matrix of the WIV estimator is given by C WIV ¼ Efðn^ WIV  nÞðn^ WIV  nÞT g T

ð21aÞ

ðF T W 1 GÞ1 g:

G T W 1 ggT W 1 G N

T 1 1 F W G : plim N

plim

T

¼ EfðG W 1 FÞ G W 1 ggT W 1 G 1

for every 40: Applying Slutsky’s theorem to (25) gives [9]

T 1 1 1 G W F C WIV ¼ plim plim N N

ð21bÞ

In what follows, we will prove that n^ WIV is asymptotically efficient, i.e., C WIV approaches the Cramer–Rao lower bound (CRLB) as N ! 1; under the assumption of small bearing noise. The

ð26Þ

Since the bearing noise is i.i.d. Gaussian, the strong law of large numbers (the pointwise-ergodic theorem [8]) implies that the sample covariance

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matrix G T W 1 F=N almost surely converges to EfG T W 1 F=Ng as N ! 1: Thus we have

T 1 1  T 1 1 G W F G W F plim ¼E : N N

Similarly, we have

T 1 1 F W G N ðF T W 1 F o Þ1 :  plim N Efcos nk g o o (32)

Under the assumption of small s2n ; the bias compensated pseudolinear estimator, which is used to construct the IV matrix G and the weighting matrix W; has a small bias [6]. Furthermore, as N ! 1; the variance of bearing angle estimates y^ k and range estimates d^k vanishes. Thus, for small bearing noise, as N ! 1 the weighting matrix can be approximated by W  W o and the IV matrix by G  F o : This leads to

T 1 1  T 1 1 G W F F W F plim E o o : (27) N N

The strong law of large numbers implies that G T W 1 ggT W 1 G=N almost surely converges to EfG T W 1 ggT W 1 G=Ng as N ! 1: Thus, the third probability limit in (26) can be written as  T 1 T 1  G T W 1 ggT W 1 G G W gg W G ¼E plim : N N

The correlation matrix EfF To W 1 o F=Ng can be written as  T 1  1 X F W F 1N 1 T E o o M mk EfuTk gM k ; (28) ¼ N N k¼0 d 2k k where EfuTk g ¼ ½Efsin y~ k g; Efcos y~ k g

(29a)

Under the small bearing noise assumption, we have  T G T W 1 ggT W 1 G gg plim  F To W 1 E W 1 o o Fo N N (33a) s2n T 1 F W F o; (33b) N o o where we have used the approximation EfggT g  s2n W o which is valid for small s2n : Substituting (31), (32) and (33b) into (26), we get 

C WIV 

s2n 1 ðF To W 1 o F oÞ : Efcos nk g2

(34)

¼ ½Efcos nk g sin yk ; Efcos nk g cos yk (29b)

Retaining the first two terms in the Taylor series expansion of cos nk under the small s2n assumption, Efcos nk g can be approximated by

¼ Efcos nk gmTk :

Efcos nk g  1  12s2n :

(29c)

Using (29c) in (28), we obtain  T 1  1 X F W F 1N 1 E o o Efcos nk gM Tk m k m Tk M k ¼ N N k¼0 d 2k

Using the above approximation, (34) can be rewritten as C WIV 

s2n ðF T W 1 F o Þ1 1  s2n þ 14 s4n o o

(35a)

(30a) 1  s2n ðF To W 1 o F oÞ :

¼

Efcos nk g T 1 F o W o F o: N

(30b)

Thus, (27) becomes

plim

G T W 1 F N

1 

N ðF T W 1 F o Þ1 : Efcos nk g o o (31)

(35b)

Thus, we have shown that under the small bearing noise assumption, the asymptotic error covariance matrix of the WIV estimator tends to the CRLB in (22), which concludes the proof of asymptotic efficiency. In general, asymptotic efficiency is preserved as long as (i) the IV matrix G tends to the noise-free measurement matrix F o ; and (ii) the weighting

ARTICLE IN PRESS K. Dog˘anc- ay / Signal Processing 85 (2005) 481–490

matrix W approximates W o up to a proportionality factor, i.e., W  aW o where a is a positive constant. If the estimate used to construct G and W satisfies the aforementioned conditions, asymptotic efficiency is guaranteed for small s2n : Referring to (35b), we see that large values of s2n have the effect of increasing the estimation covariance with respect to the CRLB.

PLE MIV WIV ML

35 30

Norm of bias estimate

488

25 20 15 10

5. Simulation examples

5 0

0.5

1

1.5

Bearing noise standard deviation (degrees)

Fig. 3. Estimation bias (N ¼ 45).

PLE MIV WIV ML CRLB

3

10

MSE estimate

This section demonstrates the asymptotic efficiency of the WIV estimator by way of simulations. The simulated target tracking geometry is depicted in Fig. 2. The initial and final bearing measurement time instants are denoted by t0 and tN1 in the figure. The measurement time interval length is fixed at T ¼ 22:50: The observer trajectory consists of three constant-velocity legs. This achieves the required observer maneuver to make the target observable. The observer collects N bearing measurements starting at r0 ¼ ½0; 0 T : The target is assumed to move at a constant acceleration with motion parameters p0 ¼ ½50; 150 T ; v0 ¼ ½2; 8 T and a ¼ ½0:5; 1:5 T : For performance comparison purposes, we use the bias norm ^  nk2 ; where n^ is an estimate of n; and the kEfng MSE which is given by Efkn^  nk22 g; i.e., the trace ^ The ML of the error covariance matrix of n: estimator was implemented using the GN algo-

2

10

−0.5

10

10

−0.3

10

−0.1

0.1

10

Bearing noise standard deviation (degrees) 350

Observer trajectory Target trajectory

tN−1

Fig. 4. Plot of MSE and CRLB (N ¼ 45).

300

y-axis

250 200 t0

150 100 50

tN−1 t0

0 -250

-200

-150

-100

-50

0

50

100

150

x-axis

Fig. 2. Simulated target tracking geometry.

200

rithm in (7). Both the GN and MIV estimators were initialized to the true target motion parameters and allowed to run for 10 iterations. To estimate the bias and MSE, 10,000 Monte Carlo simulations were performed. Figs. 3 and 4 plot the estimated bias and estimated MSE of the ML, pseudolinear (PLE), MIV and WIV estimators versus bearing noise standard deviation sn 2 f0:3 ; 0:6 ; 0:9 ; 1:2 ; 1:5 g for N ¼ 45 bearing measurements in the time interval ½0; TÞ: Fig. 4 also includes the CRLB for comparison purposes. Figs. 5 and 6 plot the

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Norm of bias estimate

6. Conclusion

PLE MIV WIV ML

35 30 25 20 15 10 5 0

489

0.5

1

1.5

Bearing noise standard deviation (degrees)

The asymptotic efficiency of the WIV estimator has been established under the assumption of small bearing noise variance. This makes the WIV estimator even more attractive given its closedform formulation which avoids the convergence difficulties associated with iterative ML estimators. In the development of the algorithm, a constant-acceleration target motion model was used rather than the traditional constant-velocity target motion model, which permits less restrictive application for the closed-form estimator. The future work will involve the development of recursive algorithm implementations.

Fig. 5. Estimation bias (N ¼ 90).

References PLE MIV WIV ML CRLB

3

MSE estimate

10

102

10−0.5

10−0.3

10−0.1

100.1

Bearing noise standard deviation (degrees)

Fig. 6. Plot of MSE and CRLB (N ¼ 90).

estimated bias and estimated MSE for N ¼ 90 bearing measurements, respectively. A comparison of Figs. 3 and 5 reveals a significant reduction in the bias of the WIV, MIV and ML estimators as N increases. This confirms the asymptotic unbiasedness of the WIV estimator. The WIV, MIV and ML estimators achieve MSE performance very close to the CRLB. As can be seen from Figs. 4 and 6, the MSE of the WIV estimator gets closer to the CRLB as N increases. This is in agreement with the asymptotic efficiency of the WIV estimator which was established in Section 4.

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