Single-platform passive emitter localization with bearing and Doppler-shift measurements using pseudolinear estimation techniques

Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Contents lists available at ScienceDirect

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Signal Processing

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journal homepage: www.elsevier.com/locate/sigpro

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Single-platform passive emitter localization with bearing and Doppler-shift measurements using pseudolinear estimation techniques

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Ngoc Hung Nguyen a,n, Kutluyıl Doğançay b,1

Q2

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a Institute for Telecommunications Research, School of Information Technology and Mathematical Sciences, University of South Australia, Mawson Lakes, SA 5095, Australia b School of Engineering, University of South Australia, Mawson Lakes, SA 5095, Australia

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a r t i c l e i n f o

abstract

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Article history: Received 27 September 2015 Received in revised form 16 December 2015 Accepted 31 January 2016

The maximum-likelihood (ML) estimator for single-platform Doppler-bearing emitter localization does not admit a closed-form solution and must be implemented using computationally demanding iterative numerical search algorithms. The iterative ML solution is vulnerable to convergence problems due to the nonconvex nature of the ML cost function and the threshold effect. To alleviate these problems, this paper presents new closed-form Doppler-bearing emitter localization algorithms in the 2D-plane based on pseudolinear estimation techniques; namely, the pseudolinear estimator (PLE), the bias-compensated PLE and the weighted instrumental variable (WIV) estimator. The biascompensated PLE aims to remove the instantaneous estimation bias inherent in the PLE. The WIV estimator incorporates the bias-compensated PLE to achieve an asymptotically unbiased estimate of the emitter position. The proposed WIV estimator is proved to be asymptotically efficient for sufficiently small measurement noise. Through simulation examples its performance is shown to be almost identical to that of the ML estimator, exhibiting small bias and approaching the Cramer–Rao lower bound at moderate noise levels. & 2016 Published by Elsevier B.V.

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Keywords: Emitter localization Bearing angle Doppler shift Pseudolinear estimate Maximum likelihood estimate Instrumental variables

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63 43 65 45

1. Introduction

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Passive emitter localization has found a variety of applications in both civilian and military domains such as user and asset localization, satellite geolocation, search and rescue, radar and sonar [1–11]. The objective of passive emitter localization is to estimate the location of a stationary emitter by utilizing passive measurements collected by a moving sensor or a number of spatially distributed

49 51 53 55 57 59

n

Corresponding author. E-mail addresses: [email protected] (N.H. Nguyen), [email protected] (K. Doğançay). 1 EURASIP member.

sensors at fixed locations. The passive sensor measurements usually include a combination of angle-of-arrival (bearing angle), time-difference-of-arrival (TDOA) and Doppler-frequency-shift data. In this paper, we focus our attention on the problem of passive Doppler-bearing emitter localization in the 2D-plane using a single moving sensor platform. Stationary emitter localization using a single moving sensor with bearing-only measurements or Doppler-only measurements has been well investigated in the literature (see e.g., [5–9] and the references therein). It was shown in [7] that the bearing-only method significantly differs from the Doppler-only method and the substantial differences between those two methods lead to a significant improvement in

http://dx.doi.org/10.1016/j.sigpro.2016.01.023 0165-1684/& 2016 Published by Elsevier B.V.

61 Please cite this article as: N.H. Nguyen, K. Doğançay, Single-platform passive emitter localization with bearing and Doppler-shift measurements using pseudolinear estimation techniques, Signal Processing (2016), http://dx.doi.org/ 10.1016/j.sigpro.2016.01.023i

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N.H. Nguyen, K. Doğançay / Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

localization performance if both bearing and Doppler measurements are utilized in the localization process. The maximum likelihood (ML) estimator for the Doppler-bearing emitter localization was presented in [7]. Despite the fact that the ML estimator is asymptotically unbiased and efficient, it does not have a closed-form solution and is often implemented as a computationally demanding iterative numerical search algorithm. The iterative ML solution is known to be vulnerable to convergence problems because of the nonconvex nature of the ML cost function [1–3,12–17]. In particular, the iterative ML algorithms require the use of an initial guess that must be sufficiently close to the actual emitter position in order to avoid divergence. In addition, the ML algorithms suffer from the threshold effect leading to divergence problems in the presence of large measurement noise. An analysis of the convergence properties of iterative algorithms can be found in [18]. The pseudolinear estimator (PLE)—a linear least squares estimator with closed-form solution—is an attractive alternative to alleviate the computational complexity and convergence problems of the iterative ML estimator [1– 3,12–17]. The PLE for bearing-only emitter localization was proposed in [8,19]. Despite its stability and low computational complexity, the PLE suffers from severe bias problems due to the correlation between the measurement matrix and the pseudolinear noise vector [8,14,19–21]. In [8], the asymptotic bias of the bearing-only PLE was characterized and a non-iterative weighted instrumental variable (WIV) estimator was proposed to overcome the PLE bias problem. The PLE approach was also applied to Doppler-only emitter localization in [9]. However, the linearization of the nonlinear Doppler equation in [9] requires the sensor to follow a particular trajectory with non-maneuvering segments. In addition, the bias problem associated with the PLE was not studied analytically. In the broader context of target localization and tracking using different sensor types, several closed-form pseudolinear algorithms have been proposed in the literature; see, e.g., [12] for TDOA localization, [13] for time-of-arrival (TOA) localization, [15] for multistatic TOA localization, [14,16] for bearing-only target motion analysis (bearing-only TMA), [3] for hybrid TDOA and bearing localization, and [17] for hybrid TDOA and frequency-difference-of-arrival localization. In contrast to the bearing-only and Doppler-only emitter localization problems, to the best of our knowledge, no closed-form solution has been proposed to date for the Doppler-bearing emitter localization problem. In [21–24] closed-form solutions were presented for the Doppler-bearing target motion analysis (DB-TMA) problem, in which the kinematic parameters (position and velocity) of a moving target are estimated from bearing and Doppler measurements obtained by a single moving sensor. However, the linearization approach applied to the nonlinear Doppler frequency equation in the DB-TMA problem cannot be readily extended to the Dopplerbearing emitter localization problem. The predominant reason for this is that, while the linearization of the Doppler equation aims to transform the Doppler information into target velocity information in the DB-TMA problem, for a stationary emitter it should instead be transformed

into emitter position information, which is not trivial. The closed-form DB-TMA algorithms can still be applied to the stationary emitter localization problem by letting them estimate the emitter velocity which is known to be zero. However, this not only increases the number of unknowns unnecessarily, but also degrades the localization performance as demonstrated in Section 8. The main objective of the paper is to develop closedform Doppler-bearing emitter localization estimators with low computational complexity and high performance advantages. The paper first presents a new Dopplerbearing PLE in which a new method of linearization for the nonlinear Doppler frequency shift equation is proposed. The proposed Doppler linearization, as different from [21–24], transforms the Doppler information into emitter position information. In addition, it does not require the sensor to follow non-maneuvering trajectory paths as in [9]. In common with the bearing-only PLE, a major disadvantage of the proposed Doppler-bearing PLE is its asymptotically nonvanishing bias caused by the correlation between the measurement matrix and the pseudolinear noise vector. To overcome this bias problem, the asymptotic bias of the proposed Doppler-bearing PLE is analysed and a bias-compensated Doppler-bearing PLE is developed based on instantaneous bias estimation for the PLE. The asymptotically unbiased Doppler-bearing WIV estimator is developed with the instrumental variable matrix and weighting matrix constructed from the biascompensated PLE. The proposed Doppler-bearing WIV estimator is also analytically shown to be asymptotically efficient, i.e., achieving the Cramér–Rao lower bound (CRLB), at moderately low measurement noise levels. The paper is organized as follows. Section 2 introduces the single-platform Doppler-bearing emitter localization problem. An overview of the ML estimator and the expression of CRLB are provided in Section 3. Section 4 presents the proposed Doppler-bearing PLE with the new linearization of Doppler frequency shift equation. In Section 5, an asymptotic bias analysis for the Doppler-bearing PLE is provided and the bias-compensated Doppler-bearing PLE is developed. Section 6 develops the asymptotically unbiased Doppler-bearing WIV estimator and presents an analysis for its asymptotic efficiency. The computational requirements of the estimators are analysed in Section 7. Comparative simulation studies are presented in Section 8 and conclusions are drawn in Section 9.

63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109

2. Problem statement and assumptions

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The problem of two-dimensional (2-D) single-platform passive emitter localization using bearing (azimuth) angle and Doppler frequency shift measurements is depicted in Fig. 1, where p ¼ ½px ; py T is the unknown position of a stationary emitter, rk ¼ ½r x;k ; r y;k T is the sensor position and vk ¼ ½vx;k ; vy;k T is the sensor velocity at time instant k A f1; …; Ng. Here the superscript T denotes the matrix transpose operator. Note that the 2-D scenario is commonly used in the literature as an approximation of the actual three-dimensional geometry for low elevationangle emitters. The objective of emitter localization is to

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Please cite this article as: N.H. Nguyen, K. Doğançay, Single-platform passive emitter localization with bearing and Doppler-shift measurements using pseudolinear estimation techniques, Signal Processing (2016), http://dx.doi.org/ 10.1016/j.sigpro.2016.01.023i

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N.H. Nguyen, K. Doğançay / Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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noise variances can vary with k and are assumed to be known a priori for bias compensation and weighting matrix computation. In practice, the assumption of Gaussian noise for bearing and Doppler-shift measurements is approximately valid for small measurement errors. The measurement errors are assumed to be sufficiently small so that the nonlinear effects inherent in bearing and Doppler measurement processes can be neglected. In this paper, we also assume that the sensor position and velocity, rk and vk , are known with negligible error.

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13 15 Fig. 1. Two-dimensional Doppler-bearing emitter localization geometry.

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estimate the emitter position p from N noisy bearing measurements and N noisy Doppler-shift measurements taken by the sensor at time instants k ¼ 1; …; N. The bearing angle θk and Doppler frequency shift Δf k at time instant k are given by the following functionals of p A R2 , rk A R2 , vk A R2 and f o A R þ (the emitted signal frequency): θk ¼ tan  1

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py  r y;k ; px  r x;k

 π oθk rπ

ð1Þ

T

f o ðrk  pÞ vk ; c J rk  p J

 Δf max;k oΔf k o Δf max;k

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Δf k ¼

31

where tan  1 is the 4-quadrant arctangent, c is the speed of signal propagation, J  J denotes the Euclidean norm, and Δf max;k ¼ ðf o =cÞ J vk J is the maximum possible frequency shift at time instant k. The magnitude and sign of the Doppler frequency shift Δf k depend on the radial projection of the sensor velocity vector with respect to the emitter and whether it points towards or away from the emitter. We assume prior knowledge of the emitted signal frequency. In practice, the emitter frequency can be estimated a priori during the loiter mode, in which the sensor position holds unchanged if the sensor is an unmanned ground vehicle or a rotary-wing unmanned aerial vehicle (e.g., quad-rotor rotorcraft), or it can be estimated by another stationary sensor in a cooperative mission. Normalizing Δf k in (2) by the factor of f o =c, the Doppler frequency shift becomes

33 35 37 39 41 43 45

ð2Þ

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where ð6aÞ

ψðpÞ ¼ ½θ1 ðpÞ; ζ 1 ðpÞ; θ2 ðpÞ; ζ 2 ðpÞ; …; θN ðpÞ; ζ N ðpÞT :

ð6bÞ

Here j  j denotes matrix determinant, ψ~ is the 2N  1 vector of noisy bearing and Doppler-shift measurements, ψðpÞ is the 2N  1 vector of noiseless bearing angles and Doppler frequency shifts as a function of p, and K ¼ diagðK1 ; K2 ; …; KN Þ is the 2N  2N diagonal covariance matrix of the sensor measurement noise vector n ¼ ½nT1 ; nT2 ; …; nTN T where nk ¼ ½nθ;k ; nζ;k T is the measurement noise vector at time instant k with the covariance matrix of Kk ¼ diagðσ 2θ;k ; σ 2ζ;k Þ. The ML estimate of the emitter position p^ ML is obtained by maximizing the log-likelihood function ln pðψ~ jpÞ over p, which is equivalent to p^ ML ¼ arg min J ML ðpÞ

83

where J ML ðpÞ is the ML cost function:

93 95 97 99 101

105 ð8Þ

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ζ~ k ¼ ζ k þnζ;k

ð4bÞ

^ ^ þ 1Þ ¼ pðiÞ ^ þ ðJT ðiÞK  1 JðiÞÞ  1 JT ðiÞK  1 ðψ~  ψðpðiÞÞÞ; pði

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where θ~ k and ζ~ k are the noisy bearing and Doppler-shift measurements, respectively, at time instant k, and nθ;k and nζ;k are the sensor measurement noises. We assume that nθ;k and nζ;k are zero-mean independent Gaussian random variables with variances σ 2θ;k and σ 2ζ;k , respectively. The

i ¼ 0; 1; …; if

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p A R2

J ML ðpÞ ¼ 12 ðψ~ ψðpÞÞT K  1 ðψ~  ψ ðpÞÞ:

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ð7Þ

ð4aÞ

ð3Þ

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θ~ k ¼ θk þnθ;k

ðr  pÞT vk ζk ¼ k : J rk  p J

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ψ~ ¼ ½θ~ 1 ; ζ~ 1 ; θ~ 2 ; ζ~ 2 ; …; θ~ N ; ζ~ N T

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The likelihood function for the bearing and Dopplershift measurements is a multivariate Gaussian probability density function:   
1 1 T 1 ~ ~ ð ψ  ψðpÞÞ ð ψ  ψ ð p Þ Þ exp  K p ψ~
pÞ ¼ 2 ð2πÞN jKj1=2 ð5Þ

In practice, the sensor measurements are corrupted by additive noise

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3. Overview of maximum likelihood estimator

The minimization of J ML ðpÞ is a nonlinear least-squares estimation problem with no closed-form solution. A numerical ML solution can be found by employing gradient or downhill simplex-based iterative search techniques (e.g., the Gauss–Newton (GN) algorithm [25], the steepest descent algorithm [26] and the Nelder–Mead simplex algorithm [27]). In particular, the GN algorithm for numerical computation of the ML estimate p^ ML is given by the following iterations:

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19

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107 109 111 113 115 117

ð9Þ

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where JðiÞ is the 2N  2 Jacobian matrix of ψðpÞ with ^ respect to p evaluated at p ¼ pðiÞ:

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JðiÞ ¼ ½JT1 ðiÞ; JT2 ðiÞ; …; JTN ðiÞT

123

ð10Þ

Please cite this article as: N.H. Nguyen, K. Doğançay, Single-platform passive emitter localization with bearing and Doppler-shift measurements using pseudolinear estimation techniques, Signal Processing (2016), http://dx.doi.org/ 10.1016/j.sigpro.2016.01.023i

N.H. Nguyen, K. Doğançay / Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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1

and

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Jk ðiÞ ¼ ½JTθ;k ðiÞ; JTζ;k ðiÞT :

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The expressions for Jθ;k ðiÞ and Jζ;k ðiÞ are given by

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Jθ;k ðiÞ ¼

^ ^ ½  sin θk ðpðiÞÞ; cos θk ðpðiÞÞ ^  rk J J pðiÞ

9 11

ð11Þ

h Jζ;k ðiÞ ¼

 vx;k

ð12Þ

p ¼ ½px ; py T and vp ¼ ½vp;x ; vp;y T denote the target position and velocity, respectively, and r ¼ ½r x ; r y T and vr ¼ ½vr;x ; vr;y T denote the sensor position and velocity at the same time instant, respectively. For convenience, we drop the time index k in this subsection. It is also noted that, to distinguish the sensor velocity from the emitter velocity, vr is used only in this subsection as the notation for the

       i sin θk p^ ðiÞ þ 12vy;k sin 2θk p^ ðiÞ ; vy;k cos 2 θk p^ ðiÞ þ 12vx;k sin 2θk p^ ðiÞ

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2

^ rk J J pðiÞ

:

ð13Þ

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13 15

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sensor velocity, whereas the notation of the sensor velocity throughout the whole paper is vk with the subscript k representing the time index. The Doppler frequency shift, after normalized by f o =c, is

23

Iterative numerical search algorithms such as the GN iterations in (9) are computationally expensive with potential divergence problems. To ensure convergence, an appropriate initialization is required. The ML estimator is known to be asymptotically unbiased and efficient. Therefore, it is commonly used as a benchmark for the purpose of performance comparison. The CRLB for the Doppler-bearing emitter localization is given by

To obtain the pseudolinear Doppler-shift equation, (17) is rewritten as

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CRLB ¼ ðJTo K  1 Jo Þ  1

ζ ¼ ðvp;x  vr;x Þ cos θ þ ðvp;y vr;y Þ sin θ

87

17 19 21

27

ð14Þ

where Jo is the Jacobian matrix evaluated at the true emitter position p.

29 4. Pseudolinear estimator 31 4.1. Linearization of bearing measurement equation 33 35 37 39

The bearing measurement equation can be linearized using the orthogonal vector sum relationship between the true bearing vector and the noisy bearing vector (see, e.g., [8] for more details), which transforms (1) and (4a) into the pseudolinear bearing equation Aθ;k p ¼ bθ;k þηθ;k

ð15Þ

41

where

43

Aθ;k ¼ ½ sin θ~ k ;  cos θ~ k 

ð16aÞ

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bθ;k ¼ ½ sin θ~ k ;  cos θ~ k rk

ð16bÞ

ηθ;k ¼ J p  rk J sin nθ;k :

ð16cÞ

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4.2. Linearization of Doppler-shift measurement equation

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In this subsection we first explain why the linearization of the nonlinear Doppler-shift measurement equation in the DB-TMA problem [21–24] cannot be applied to Doppler-bearing emitter localization. We then propose a new linearization approach for the Doppler-shift measurement equation.

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4.2.1. Doppler-shift pseudolinear equation in DB-TMA In the DB-TMA problem, the position and velocity of a moving target are estimated from bearing and Doppler measurements collected by a single moving sensor. Let

ðp rÞT ðvp vr Þ : ζ¼ J p r J

79 81

ð17Þ 83

ð18Þ

1

where θ ¼ tan ððpy r y Þ=ðpx  r x ÞÞ is the bearing angle. Writing (18) in terms of the target motion parameter vector ½px ; py ; vp;x ; vp;y T , the pseudolinear Doppler-shift equation becomes 2 3 px " # 6 p 7 vr;x 6 y 7 7 ¼ ζ þ½ cos θ sin θ : ½0 0 cos θ sin θ6 6 vp;x 7 vr;y 4 5

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vp;y

ð19Þ

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Eq. (19) reveals that the Doppler linearization in the DB-TMA problem transforms the Doppler information into the target velocity information as the target position does not directly contribute to the pseudolinear Doppler-shift equation (note that px and py are multiplied with zero in (19)). In emitter localization, the emitter is assumed stationary. As a result, substituting vp;x ¼ vp;y ¼ 0 into (19) makes the left-hand side term of (19) zero eliminating all target motion parameters. Consequently, (19) cannot be used for Doppler-bearing emitter localization. In the next subsection, we propose a new linearization approach for the Doppler-frequency-shift equation, which transforms the Doppler information into the emitter position information.

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4.2.2. New pseudolinear Doppler-shift equation for stationary emitter For a stationary emitter the Doppler frequency shift in (3) is ζ k ¼  vx;k cos θk  vy;k sin θk : Substituting sin

p r θk ¼ py  ry;k x;k x

ð20Þ

cos θk into (20) gives

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ðζ k þ vx;k cos θk Þpx þ ðvy;k cos θk Þpy ¼ ζ k r x;k þ ðvx;k r x;k þ vy;k r y;k Þ cos θk

115

ð21Þ

Please cite this article as: N.H. Nguyen, K. Doğançay, Single-platform passive emitter localization with bearing and Doppler-shift measurements using pseudolinear estimation techniques, Signal Processing (2016), http://dx.doi.org/ 10.1016/j.sigpro.2016.01.023i

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p r

and substituting cos θk ¼ px  rx;k sin θk into (20) gives y;k

as

ðvx;k sin θk Þpx þðζk þ vy;k sin θk Þpy ¼ ζk r y;k þ ðvx;k r x;k þvy;k r y;k Þ sin θk :

p^ PLE ¼ p  ðA AÞ

y

ð22Þ Adding (21) and (22), we obtain     ζ k þ vx;k ð sin θk þ cos θk Þ px þ ζ k þvy;k ð sin θk þ cos θk Þ py ¼ ζ k ðr x;k þ r y;k Þ þ ðvx;k r x;k þ vy;k r y;k Þð sin θk þ cos θk Þ:

ð23Þ

Replacing the noiseless bearing angle θk and Doppler-shift ζk by noisy bearing and Doppler-shift measurements θ~ k and ζ~ k , respectively, yields the pseudolinear equation for Doppler-shift measurements: Aζ;k p ¼ bζ;k þ ηζ;k

17

Aζ;k ¼ ζ~ k uT1 þð sin θ~ k þ cos θ~ k ÞvTk

ð25aÞ

19

bζ;k ¼ ζ~ k uT1 rk þ ð sin θ~ k þ cos θ~ k ÞvTk rk

ð25bÞ

21

 ηζ;k ¼ uT1 ðp rk Þnζ;k þ vTk ðp  rk Þ ð cos θk  sin θk Þ sin nθ;k



2ð sin θk þ cos θk Þ sin ðnθ;k =2Þ : 2

ð25cÞ

Here the vector u1 is defined by u1 ¼ ½1; 1 and the derivation of the pseudolinear noise ηζ;k can be found in Appendix A.

27 4.3. Closed-form pseudolinear estimator 29 31 33 35 37 39 41 43

Stacking the pseudolinear Eqs. (15) and (24) for the bearing and Doppler-shift measurements at time instant k yields Ak p ¼ bk þ ηk

ð26Þ

where Ak ¼ ½ATθ;k ; ATζ;k T , bk ¼ ½bθ;k ; bζ;k T , and ηk ¼ ½ηθ;k ; ηζ;k T . Concatenating (26) for k ¼ 1; …; N, we have Ap ¼ bþ η

¼ ½AT1 ; AT2 ; …; ATN T

b¼

69

CPLE ¼ Efðp^ PLE  pÞðp^ PLE pÞ g ¼ EfðA AÞ T

T

ð31Þ 1

T

T

T

A ηη AðA AÞ

1

g:

ð32Þ

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5. Bias compensation for the pseudolinear estimator

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5.1. Bias of the pseudolinear estimator

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The pseudolinear estimate p^ PLE asymptotically converges in probability to

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p

p^ PLE ⟶plimfp^ PLE g as N-1

ð28aÞ

T T T ½b1 ; b2 ; …; bN T

ð28bÞ

( )1 ( ) AT A AT η plimfp^ PLE g ¼ p  E E : N N

p^ PLE ¼ ðAT AÞ  1 AT b

53

The target is observable if ðAT AÞ is invertible, i.e., the A matrix is full-rank. For the Doppler-bearing emitter localization problem, A becomes rank-deficient only if the sensor is moving radially towards or away from the emitter during the entire observation interval. However, such a special sensor-emitter geometry rarely occurs in practice. Using (27) and (29b), the pseudolinear estimate p^ PLE can be written in terms of the pseudolinear noise vector η

ð29bÞ

87

91 93

( ) ( T ) ( T ) ( ) N N N X X X Aζ;k ηζ;k Aθ;k ηθ;k AT η ATk ηk ¼ þ ¼ E E E E N N N N k¼1 k¼1 k¼1

97 ð35Þ

where ( T ) N N X Aθ;k ηθ;k 1X E Ef sin 2 nθ;k gðp rk Þ ¼ N N k¼1 k¼1 N X

E

k¼1

95

99 101

ð36aÞ

( T ) N Aζ;k ηζ;k 1X ¼ Efn2ζ;k guT1 ðp rk Þu1 N Nk¼1

p^ PLE ¼ arg min J Ap b J

51

85

Here the convergence in probability is defined by [28]: p XðNÞ⟶X as N-1 iff PfjXðNÞ  X j4 ϵg-0 as N-1 for every ϵ 40, where X is referred as the probability limit of XðNÞ and is commonly denoted by plimfXðNÞg. From (26) and (27), we have

49

ð29aÞ

ð34aÞ

89

þ

p A R2

81

ð34bÞ

47

ð28cÞ

2

ð33Þ

83

( )1 ( ) AT A AT η plim plimfp^ PLE g ¼ p  plim N N



61

65 67

The Doppler-bearing pseudolinear estimate p^ PLE , i.e., the least-squares solution of (27), is given by

59

ð30Þ

δPLE ¼ Efp^ PLE g p ¼  EfðAT AÞ  1 AT ηg

η ¼ ½ηT1 ; ηT2 ; …; ηTN T :

57

A η:

The bias and error covariance matrix of p^ PLE are therefore given by

45

55

T

ð27Þ

where A

1

where

T

25

63 T

ð24Þ

where

23

5

103 105

N n n o 2X ζ ð sin θk þ cos θk ÞE sin 2 θ;k vTk ðp  rk Þu1 2 Nk¼1 k

N n  n n o 1X ð 2ð sin θk þ cos θk Þ2 E sin 2 θ;k 2 Nk¼1 n n o  2E sin 4 θ;k 2 n o 2 T þ ð cos θk  sin θk Þ2 E sin nθ;k vk ðp  rk Þvk g:

ð36bÞ The crosscorrelation vectors in (36b) are derived from (16a), (16c), (25a) and (25c) after some algebraic manipulations. They are given in terms of noiseless bearing angles θk, noiseless Doppler shifts ζk, bearing noise nθ;k and Doppler-shift noise nζ;k . We can see from (35) and (36) that EfAT η=Ng a0 even for N-1, implying that the crosscorrelation between the

Please cite this article as: N.H. Nguyen, K. Doğançay, Single-platform passive emitter localization with bearing and Doppler-shift measurements using pseudolinear estimation techniques, Signal Processing (2016), http://dx.doi.org/ 10.1016/j.sigpro.2016.01.023i

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1 3 5 7 9

entries of A and η does not vanish as N-1. Convergence in probability implies convergence in distribution [28]. Thus, from (33) we have ( )1 ( ) AT A AT η d ^ p PLE ⟶p  E E as N-1 ð37Þ N N where the convergence in distribution is defined by [28] as: d

11 13 15 17 19





XðNÞ⟶X as N-1 iff FfXðNÞg-FfX g as N-1: Here Ffg denotes the distribution function. Eq. (37) is equivalent to ( )1 ( ) AT A AT η E Efp^ PLE g-p  E as N-1: ð38Þ N N Thus we have ( )1 ( ) AT A AT η as N-1 E δPLE - E N N

25 27 29 31 33 35

ð39Þ

37 39 41 43 45 47 49 51 53

which confirms the asymptotically nonvanishing bias of the PLE. For sufficiently large N, the pseudolinear estimation bias can be approximated by ( )1 ( ) AT A AT η δPLE  E E : ð40Þ N N

5.2. Bias-compensated pseudolinear estimator The bias-compensated PLE is developed by estimating the instantaneous bias δins of the PLE from the available sensor measurements. From (30), δins is given by δins ¼ ðAT AÞ  1 AT η

ð41Þ

which cannot be computed because the pseudolinear noise η is unknown. However, the instantaneous bias can be approximated as δ^ ins ¼  ðAT AÞ  1 EfAT ηg ð42Þ n o n o where EfAT ηg ¼ NE AT η=N with E AT η=N given in (35). n o Since E AT η=N is a function of the unknown noiseless bearing angles θk and Doppler shifts ζk, the approximate instantaneous bias δ^ ins cannot be computed either. An estimate of δ^ ins can be obtained by replacing θk and ζk by their estimates θ^ k and ζ^ k computed from the pseudolinear estimate p^ PLE :

^ ^ T ηg δ^ ins ¼  ðAT AÞ  1 EfA

ð43Þ

where

55 57

Here

61

nθ;k g  σ 2θ;k ,

Ef sin 2 ðnθ;k =2Þg  σ 2θ;k =4 and Ef sin 4 ðnθ;k =2Þg  0 in (36b). The bias-compensated Doppler-bearing PLE is now obtained by subtracting the estimated instantaneous bias ^ δ^ ins from the pseudolinear estimate p^ PLE : ^ T ηg: p^ BCPLE ¼ p^ PLE þ ðAT AÞ  1 EfA

63 65 67 69

ð45Þ

This bias-compensated pseudolinear estimator is not strictly unbiased since the exact instantaneous bias cannot be computed and the estimated instantaneous bias obtained from the available sensor measurements is employed instead. Nevertheless, its bias is significantly smaller than the bias of the pseudolinear estimator.

71 73 75 77

6. Asymptotically unbiased weighted instrumental variable estimator

81

6.1. The WIV estimator

83

The bias-compensated Doppler-bearing PLE presented in Section 5.2 still suffers from bias problems particularly in large-noise scenarios even though it can lead to a significant reduction in the estimation bias compared to the PLE. In this subsection, we develop a Doppler-bearing WIV estimator which enjoys asymptotic unbiasedness. To construct an instrumental variable (IV) matrix and weighting matrix for the WIV estimator we will use the biascompensated PLE in (45). The bias of the PLE, arising from the correlation between A and η, can be eliminated by modifying the PLE normal equations from AT Ap^ PLE ¼ AT b to GT Ap^ IV ¼ GT b [8,14]. The resulting IV estimator is therefore given by

85

p^ IV ¼ ðGT AÞ  1 GT b:

ð46Þ

Here G is the IV matrix and must be constructed such that EfGT A=Ng is nonsingular and EfGT η=Ng ¼ 0 as N-1 to ensure the consistency, i.e., plimfp^ IV g ¼ p as N-1, and the asymptotic unbiasedness, i.e., Efp^ IV g ¼ p as N-1, of the IV estimator p^ IV [29]. The optimal choice for the IV matrix is the noise-free version of the measurement matrix A [8]. However, this optimal IV matrix is not available as it is a function of the unknown noiseless bearing angles θk and Doppler shifts ζk. A suboptimal IV matrix can be obtained by replacing the unknown θk and ζk with their estimates θ^ k and ζ^ k computed from the biascompensated PLE p^ BCPLE : G ¼ ½GT1 ; GT2 ; …; GTN T

ð47Þ

87 89 91 93 95 97 99 101 103 105 107 109 111 113

where

( T ) ( T ) N N n o X X Aζ;k ηζ;k Aθ;k ηθ;k E^ E^ E^ AT η ¼ N þN : N N k¼1 k¼1

59

we make the following approximations: Ef sin

2

79

21 23

and ζ^ k computed from p^ PLE . For small measurement noise,

 T  P ^ Aζ;k ηζ;k have the same and N k¼1E N  T   T  PN PN Aζ;k ηζ;k Aθ;k ηθ;k and in expressions as k¼1E k¼1E N N PN

k¼1

E^



ð44Þ

ATθ;k ηθ;k N



(36b), respectively, except that θk and ζk are replaced by θ^ k

Gk ¼ ½GTθ;k ; GTζ;k T

ð48Þ

and

115 117

Gθ;k ¼ ½ sin θ^ k ;  cos θ^ k 

ð49aÞ

Gζ;k ¼ ζ^ k uT1 þ ð sin θ^ k þ cos θ^ k ÞvTk :

ð49bÞ

The IV matrix G constructed from p^ BCPLE asymptotically has no correlation with η [8,14], i.e.,

Please cite this article as: N.H. Nguyen, K. Doğançay, Single-platform passive emitter localization with bearing and Doppler-shift measurements using pseudolinear estimation techniques, Signal Processing (2016), http://dx.doi.org/ 10.1016/j.sigpro.2016.01.023i

119 121 123

N.H. Nguyen, K. Doğançay / Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1

E

( ) GT η ¼ 0 as N-0: N

ð50Þ

3 5 7 9 11

This uncorrelation can be explained by noting that the covariance of p^ BCPLE tends to zero as N-1 and thus the correlation between p^ BCPLE and η vanishes as N-1. In practice, we have EfGT η=Ng  0 for sufficiently large N. To reduce the IV estimation error covariance, the weighting matrix W, which is obtained from the biascompensated PLE p^ BCPLE , is incorporated into the IV estimation to produce a WIV estimate of the emitter position: p^ WIV ¼ ðG W T

13

1

AÞ

1

T

G W

1

ð51Þ

b:

The expression of W is given by 15

W ¼ diagðW1 ; W2 ; …; WN Þ

17

where "

W k;12 W k;22

ð52Þ

#

19

W k;11 Wk ¼ W k;21

21

and

23

W k;11 ¼ J p^ BCPLE  rk J 2 σ 2θ;k

25

W k;12 ¼ J p^ BCPLE  rk J vTk ðp^ BCPLE  rk Þð cos θ^ k  sin θ^ k Þσ 2θ;k

ð53Þ

ð54aÞ

ð54bÞ

7

Since the measurement noise vector nk is statistically independent over time with a finite covariance matrix, GTk Wk 1 Ak is also independent over time and its entries κij;k have finite variances. Moreover, we have N Efκ 2 g X ij;k k¼1

2

k PN

N n oX 1 r max Efκ 2ij;k g : 2 k k¼1

67 69

2

For small measurement noise, the bias-compensated PLE p^ BCPLE , which is used to construct G and W, has small bias. In addition, the variance of p^ BCPLE and the variances of θ^ k and ζ^ k computed from p^ BCPLE vanish as N-1. As a result, the IV matrix can be approximated by G  Ao and the weighting matrix by W  Wo , where Ao and Wo obtained from the true values of p, θk and ζk are the noiseless versions of A and W, respectively. Thus, we have ( ) ( ) GT W  1 A GTo Wo 1 A GT W  1 EfAg ATo Wo 1 Ao  : E E  o o N N N N

27

W k;21 ¼ W k;12

29

 2  2 W k;22 ¼ uT1 ðp^ BCPLE  rk Þ σ 2ζ;k þ vTk ðp^ BCPLE  rk Þð cos θ^ k  sin θ^ k Þ σ 2θ;k :

31

ð54dÞ

ð60Þ

The weighting matrix W is an approximation of the covariance matrix Σ ¼ EfηηT g of the pseudolinear noise vector η, which is derived in Appendix B. This approximation stems from replacing the true values of p and θk by their estimates of p^ BCPLE and θ^ k computed from p^ BCPLE , and using Ef sin 2 nθ;k g  σ 2θ;k and Ef sin 4 ðnθ;k =2Þg  0.

Here we use the approximation EfAg  Ao , which ignores the second and higher-order noise terms under the assumption of small measurement noise. Similarly, we have ( ) AT W  1 G p AT W  1 G ⟶E as N-1 ð61Þ N N

33 35 37

ð54cÞ

39

6.2. Asymptotic efficiency of the WIV estimator

41

In this subsection, we prove that the proposed WIV estimator is asymptotically efficient, i.e., its error covariance approaches the CRLB as N-1 under the assumption of small measurement noise. The error covariance matrix of the WIV estimator is given by

43 45 47 49 51 53

CWIV ¼ Efðp^ WIV  pÞðp^ WIV  pÞT g ¼ EfðGT W  1 AÞ  1 GT W  1 ηηT W  1 GðAT W  1 GÞ  1 g

55 57 59 61

ð56Þ Here we assume that the emitter is observable, i.e., GT W  1 A=N is nonsingular, which can be expressed as N GT W  1 A 1 X ¼ GT W  1 Ak : N Nk¼1 k k

( ) GT W  1 ηηT W  1 G p GT W  1 ηηT W  1 G ⟶E as N-1 N N (

ð57Þ

73 75 77 79 81 83 85 87 89 91 93 95 97 99

103 105

with

Let Q ¼ ðp^ WIV  pÞðp^ WIV  pÞT denote the term inside the expectation operation in (55), which can be rewritten as !1 !1 1 GT W  1 A GT W  1 ηηT W  1 G AT W  1 G Q¼ : N N N N

ð62Þ

and

E

71

101

with ( ) AT W  1 G A T W  1 Ao  o o ; E N N

ð55Þ

65

ð58Þ

converges to π 2 =6 as N-1 [30], k ¼ 1 1=k 2 2 Efκ g=k also converges as N-1. Consequently, k¼1 ij;k the Kolmogorov criterion is satisfied for each entry κij;k of GTk Wk 1 Ak . Now, applying Kolmogorov's strong law of large P T 1 numbers [31], 1=N N k ¼ 1 Gk Wk Ak almost surely conPN T 1 verges to 1=N k ¼ 1 EfGk Wk Ak g as N-1. Since almost surely convergence implies convergence in probability [28], we have ( ) GT W  1 A p GT W  1 A ⟶E as N-1: ð59Þ N N As PN

63

GT W  1 ηηT W  1 G N

) 

ATo Wo 1 EfηηT gWo 1 Ao N



ATo Wo 1 Wo Wo 1 Ao ATo Wo 1 Ao  : N N

ð63Þ

109 111 113 ð64Þ 115

Applying the rules for the convergence of functions of random sequences [28] to (56) and (59)–(64), and noting that convergence in probability implies convergence in distribution, we finally obtain p

Q ⟶plimfQ g  ðATo Wo 1 Ao Þ  1 d

as N-1

⟹Q ⟶plimfQ g  ðATo Wo 1 Ao Þ  1 as N-1:

107

ð65aÞ ð65bÞ

Please cite this article as: N.H. Nguyen, K. Doğançay, Single-platform passive emitter localization with bearing and Doppler-shift measurements using pseudolinear estimation techniques, Signal Processing (2016), http://dx.doi.org/ 10.1016/j.sigpro.2016.01.023i

117 119 121 123

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8

1

As a result, in the asymptotic sense (i.e., as N-1), we have

3

CWIV ¼ EfQ g ¼ plimfQ g

ð66aÞ

5

CWIV  ðATo Wo 1 Ao Þ  1

ð66bÞ

7

CWIV  ðJTo K  1 Jo Þ  1

ð66cÞ

9

CWIV  CRLB

ð66dÞ

11 13

where ATo Wo 1 Ao ¼ JTo K  1 Jo (see Appendix C for proof). This implies that the error covariance matrix of the WIV estimator asymptotically attains the CRLB under the assumption of small measurement noise.

15 17

7. Computational complexity

19

This section provides a detailed comparison of the computational complexities of the proposed closed-form estimators (the PLE, the bias-compensated PLE, and the WIV estimator) and the ML estimator implemented via the iterative GN algorithm. Given N bearing measurements, N Doppler-shift measurements, N sensor positions, and N sensor velocities, the total computational complexity of the PLE is 20N þ 9 multiplications and one division. Computing the estimated ^ instantaneous bias δ^ ins in (43) requires an additional 19N þ4 multiplications, N divisions and N square root operations. The total computational complexity of the bias-compensated PLE (including that of the PLE) is therefore 39N þ 13 multiplications, N þ1 divisions and N square root operations. Implementing the WIV estimator requires a total of 83N þ 23 multiplications, 3N þ 2 divisions and 3N square root operations, which includes the computational complexity of the bias-compensated PLE. The computational complexity required for each iteration of the GN algorithm is 29N þ 9 multiplications, 3N þ 1 divisions and N square root operations. If the GN algorithm is initialized to the bias-compensated PLE, the total computational complexity of the GN algorithm is ð29N þ 9Þðif þ1Þ þ ð39N þ13Þ multiplications, ð3N þ 1Þðif þ 1Þ þ ðN þ 1Þ divisions and Nðif þ 1Þ þ N square root operations. Here if þ1 is the total number of the GN iterations. The computational complexities for large N can be approximated by retaining only the complexity terms that are dependent on N. Table 1 summarizes the approximate computational complexities for all the algorithms, which confirm the computational advantage of the proposed closed-form pseudolinear algorithms over the iterative ML algorithm.

21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

sensor collects N¼40 bearing and Doppler-shift measurements at equally spaced points along the trajectory segment 5 rr x;k r35. The bearing and Doppler-shift measurements are subject to i.i.d zero-mean Gaussian noise with constant variances of σ 2θ and σ 2ζ , respectively. The values of the noise standard deviations σ θ and σ ζ used in the simulations are listed in Table 2. Note that the assumption of constant noise variances is approximately valid in the simulated scenario since the sensor-target geometry does not change much during the observation period. The ML estimator is implemented using the GN algorithm (9) which is initialized to the bias-compensated PLE and iterated 10 times. For performance comparison purposes, the bias norm and root-mean-squared-error (RMSE) of the algorithms are estimated via 100,000 Monte Carlo simulation runs. The bias norm is defined by J Efp^ X g  p J and RMSE by ðtr Efðp^ X  pÞðp^ X pÞT gÞ1=2 , where p^ X is the estimate of p obtained from the ML estimator, PLE, biascompensated PLE, or WIV estimator. The square root of the trace of the CRLB matrix (labelled in figures as CRLB for simplicity) is also computed and plotted along with the RMSE.

63 65 67 69 71 73 75 77 79 81 83

Table 1 Comparison of computational complexities. Algorithm

Multiplication

Division

Square root

PLE Bias-compensated PLE WIV GN

20N 39N 83N ð29if þ 68ÞN

– N 3N ð3if þ 4ÞN

– N 3N ðif þ 2ÞN

85 87 89 91 93 95 97 99 101 103

53

Fig. 2. Simulated emitter localization geometry.

8. Simulation examples 55 8.1. Simulation setup 57 59 61

The simulated emitter localization geometry is depicted in Fig. 2. The true emitter position is p ¼ ½70; 60T m. The sensor moves with a constant velocity of vk ¼ ½  10; 2T m=s following a linear path given by r y;k ¼  0:2r x;k þ 10. The

105 Table 2 Measurement noise standard deviation.

107 10

109

1 1.5 2 2.5 3 3.5 4 4.5 5 σ θ (degree) 0.5 σ ζ (m/s) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

111

Index

1

2

3

4

5

6

7

8

9

Please cite this article as: N.H. Nguyen, K. Doğançay, Single-platform passive emitter localization with bearing and Doppler-shift measurements using pseudolinear estimation techniques, Signal Processing (2016), http://dx.doi.org/ 10.1016/j.sigpro.2016.01.023i

N.H. Nguyen, K. Doğançay / Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1

8.2. Simulation example 1: proposed closed-form estimators versus iterative ML estimator

3 5 7 9 11 13 15 17 19 21

Fig. 3 shows the bias norm and RMSE performance of the three proposed closed-form estimators, i.e., the PLE, biascompensated PLE (BCPLE) and WIV estimators, as well as the iterative ML estimator (MLE) versus the measurement noise standard deviation in Table 2. The PLE, as expected, suffers from a severe bias. This bias is significantly reduced by employing the BCPLE. In particular, the PLE bias is almost completely removed by the BCPLE at small noise levels (σ θ r 31 and σ ζ r0:3 m=s). However, the BCPLE starts exhibiting bias from σ θ ¼ 3:51 and σ ζ ¼ 0:35 m=s and its bias increases considerably as the noise level increases. On the other hand, both the WIV and MLE have almost no bias even for large noise levels. It is observed that the MLE performs slightly worse than the proposed WIV in terms of bias performance. With reference to the RMSE performance, the BCPLE outperforms the PLE, while the WIV and MLE exhibit the best performance with their RMSE performance almost achieving the CRLB. It is noted that, although the proposed

9

WIV exhibits a comparable performance with the MLE, it requires much less computations than the MLE. Fig. 4 plots the bias norm and RMSE performance of the estimators against N, i.e., the number of times that the sensor collects measurements during the observation interval, for σ θ ¼ 51 and σ ζ ¼ 0:5 m=s (noise index 10 in Table 2). Here, the sensor measurements are collected at equally spaced points along the same trajectory segment 5 r r x;k r 35. The MLE fails to converge for N o20 while the proposed closed-form PLE, BCPLE and WIV appear to perform well with the RMSE of WIV quite close to the CRLB. It is also observed in Fig. 4 that the PLE and BCPLE exhibit nonvanishing biases as N increases while both the biases of the WIV and MLE tend to zero as N increases. This observation confirms the asymptotically nonvanishing bias of the PLE and BCPLE, as well as the asymptotic unbiasedness of the WIV and MLE. In addition, we observe that the RMSE performance of the WIV and MLE closely matches the CRLB for N Z 40, confirming the asymptotic efficiency of these two estimators.

63 65 67 69 71 73 75 77 79 81 83

23

85

25

87

27

89

29

91

31

93

33

95

35

97

37

99

39

101

41

103

43

105

45

107

47

109

49

111

51

113

53

115

55

117

57

119 121

59 61

Fig. 3. Bias norm, RMSE and CRLB versus noise standard deviation in Table 2 for the Doppler-bearing PLE, BCPLE, WIV and MLE.

Fig. 4. Bias norm, RMSE and CRLB versus N for the Doppler-bearing PLE, BCPLE, WIV and MLE.

Please cite this article as: N.H. Nguyen, K. Doğançay, Single-platform passive emitter localization with bearing and Doppler-shift measurements using pseudolinear estimation techniques, Signal Processing (2016), http://dx.doi.org/ 10.1016/j.sigpro.2016.01.023i

123

10

1

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8.3. Simulation example 2: Doppler-bearing localization versus Doppler-bearing target motion analysis

8.4. Simulation example 3: Doppler-bearing localization versus bearing-only localization

As discussed in Section 1, the closed-form DB-TMA algorithms can be applied to stationary emitter localization by estimating both the emitter velocity and emitter position. However, ignoring the prior knowledge of zero emitter velocity will greatly degrade the performance of emitter position estimation. To demonstrate this, we compare the CRLB of the DB-TMA approach (in which the prior knowledge of zero emitter velocity is ignored) with the CRLB of the Doppler-bearing emitter localization approach (in which the zero emitter velocity is explicitly incorporated by estimating the emitter position only). To ensure a fair comparison, Fig. 5 plots the square root of the sum of the first two diagonal elements of the 4  4 CRLB matrix of the DB-TMA approach, which corresponds to the lower bound for the RMSE of the position estimate, along with the square root of the trace of the 2  2 CRLB matrix of the Doppler-bearing localization approach for the simulated localization scenario. The CRLB for the DB-TMA approach is derived in Appendix D. It is observed from Fig. 5 that the CRLB of the position estimate of the DB-TMA approach is significantly larger than the CRLB of the Doppler-bearing localization approach. This confirms the performance degradation in position estimation produced by the DB-TMA approach as a consequence of ignoring the prior knowledge of zero emitter velocity. The proposed WIV estimator closely achieves the CRLB of Doppler-bearing localization as analytically shown in Section 6.2 and demonstrated via simulations in Section 8.2. As a result, the proposed WIV estimator will significantly outperform closed-form DB-TMA algorithms whose performance is bounded by their CRLB. The performance of the Dopplerbearing TMA approach is superior to that of the bearingonly TMA approach since the former approach makes use of both Doppler and bearing measurements while the latter only utilizes bearing measurements. Therefore the proposed Doppler-bearing localization algorithms will also outperform the bearing-only TMA algorithms presented in [14] for stationary target localization.

If the target velocity is set to zero in the DB-TMA problem, the pseudolinear Doppler-shift Eq. (19) cannot be employed (see Section 4.2.1 for more details). This effectively reduces the DB-TMA pseudolinear estimation problem to the pseudolinear bearings-only localization problem [8]. The Doppler-bearing localization technique is known to provide a significant improvement in localization performance compared to the bearing-only localization technique by making use of both bearing and Doppler information [7]. To demonstrate this improvement, Fig. 6 plots the performance of the closed-form bearing-only emitter localization algorithms proposed in [8] at bearing noise levels in Table 2. Note that the bearing-only WIV estimator is implemented based on the bearing-only BCPLE rather than the bearing-only PLE. It is apparent from the comparison of Figs. 3 and 6 that, by incorporating the Dopplershift measurements, the closed-form Doppler-bearing estimators significantly outperform the closed-form

65

3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41

63

67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103

43

105

45

107

47

109

49

111

51

113

53

115

55

117

57

119 121

59 61

Fig. 5. CRLB comparison between Doppler-bearing localization and DBTMA.

Fig. 6. Bias norm, RMSE and CRLB versus bearing noise standard deviation in Table 2 for the bearing-only PLE, BCPLE, WIV (compare to Fig. 3).

Please cite this article as: N.H. Nguyen, K. Doğançay, Single-platform passive emitter localization with bearing and Doppler-shift measurements using pseudolinear estimation techniques, Signal Processing (2016), http://dx.doi.org/ 10.1016/j.sigpro.2016.01.023i

123

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1 3 5 7

bearing-only estimators in terms of both bias and RMSE performance. Moreover, we also observe that the gap between the RMSE of the Doppler-bearing WIV estimator and the corresponding Doppler-bearing CRLB is much smaller than the gap between the RMSE of the bearingonly WIV estimator and the corresponding bearingonly CRLB.

9

13

In this paper we have developed three new closed-form Doppler-bearing emitter localization algorithms based on pseudolinear estimation. We proposed a new linearization method for the nonlinear Doppler-frequency-shift equation to transform the Doppler information into emitter position information, which was then employed in the development of the proposed estimators. In contrast to the iterative ML estimator which is computationally expensive and vulnerable to convergence problems, the presented estimators are closed-form with low computational complexity and inherent stability. Both the PLE and the biascompensated PLE suffer from bias problems even though the bias of the PLE is greatly reduced by the biascompensated PLE by estimating the instantaneous bias of the PLE. In contrast, the WIV estimator is asymptotically unbiased. The WIV estimator with the IV and weighting matrices constructed from the bias-compensated PLE was also analytically shown to be asymptotically efficient under the assumption of small sensor noise. Comparative simulation studies were carried out to compare the bias and RMSE performance of the proposed estimators with that of the ML estimator. The simulation results confirm that the WIV estimator outperforms the PLE and the biascompensated PLE while it exhibits an estimation performance almost identical to that of the computationally demanding iterative ML estimator.

19 21 23 25 27 29 31 33 35 37 39

45 47 49 51

Efη2θ;k g

Efηθ;k ηζ;k g

Efηζ;k ηθ;k g

Efη2ζ;k g

5

ð69Þ

Appendix A

67 Efη2θ;k g ¼ J p rk J 2 Ef sin 2 nθ;k g

ð70aÞ

Efηζ;k ηθ;k g ¼ Efηθ;k ηζ;k g 

ð70cÞ

2

 2 Efη2ζ;k g ¼ uT1 ðp  rk Þ Efn2ζ;k g þ vTk ðp  rk Þ ð4ð sin θk n n o þ cos θk Þ2 E sin 4 θ;k 2 þ ð cos θk  sin θk Þ2 Ef sin 2 nθ;k g :



þ cos ~θ k  cos θk Þ vx;k ðpx r x;k Þ  vy;k ðpy  r y;k Þ :

83

P T 1 T 1 Since ATo Wo 1 Ao ¼ N Jo ¼ k ¼ 1 A o;k W o;k A o;k and Jo K PN T 1 J K J , we will show, in what follows, that o;k k ¼ 1 o;k k 1 ATo;k Wo;k Ao;k ¼ JTo;k Kk 1 Jo;k which proves ATo Wo 1 Ao ¼ JTo K  1 Jo . Note that Ao;k , Wo;k and Jo;k are the noiseless versions of Ak in (26), Wk in (53), and Jk in (11), respectively. 1 To prove ATo;k Wo;k Ao;k ¼ JTo;k Kk 1 Jo;k , we introduce a vector η k , an approximation of ηk after removing the second and higher-order noise terms, given by " # J p rk J nθ;k ηk ¼ : ð71Þ uT1 ðp  rk Þnζ;k þ vTk ðp  rk Þð cos θk  sin θk Þnθ;k The Jacobian matrix Vk ¼ ∂η k =∂nk of η k with respect to nk is " # 0 J p rk J : ð72Þ Vk ¼ vTk ðp  rk Þð cos θk  sin θk Þ uT1 ðp rk Þ

85 87 89 91 93 95 97 99 101 103 105 107

ð73Þ

ð67Þ

109

Appendix D

111 113

55

Appendix B

57

The covariance matrix of the pseudolinear noise vector η is Σ ¼ EfηηT g ¼ diagðEfη1 ηT1 g; Efη2 ηT2 g; :::;EfηN ηTN gÞ

pk ¼ p1 þ t k vp;1 ¼ Mk ξ;

ð68Þ

75

81

Consider a DB-TMA problem where the position and velocity of a moving target are estimated from bearing and Doppler measurements collected by a single moving sensor in the 2D plane. Let pk and vp;k denote the target position and velocity, respectively, and rk and vr;k denote the sensor position and velocity at time instant k, respectively. A constant-velocity target dynamic model is commonly assumed in DB-TMA problems:

59

73

79

Appendix C

1 Ao;k : ¼ ATo;k Wo;k

53

71

ð70dÞ

¼ ðVk Jo;k ÞT ðVk KVTk Þ  1 ðVk Jo;k Þ

The expression for ηζ;k in (25c) is then obtained by substituting θ~ k ¼ θk þnθ;k and ζ~ k ¼ ζ k þ nζ;k into (67).

69

77

JTo;k Kk 1 Jo;k ¼ JTo;k VTk ðVTk Þ  1 Kk 1 Vk 1 Vk Jo;k

Subtracting (23) from (24) and rearranging yield  ηζ;k ¼ ðζ~ k  ζ k Þ ðpx  r x;k Þ þ ðpy  r y;k Þ þ ð sin θ~ k  sin θk

65

and

Through some algebraic manipulations, it is straightforward to verify that Ao;k ¼ Vk Jo;k and Wo;k ¼ Vk Kk VTk which implies

41 43

Efηk ηTk g ¼ 4

63

3

ð70bÞ

9. Conclusions

17

2

Efηθ;k ηζ;k g ¼ J p  rk J vTk ðp rk Þð cos θk  sin θk ÞEf sin 2 nθ;k g

11

15

where

11

k ¼ 1; …; N

ð74aÞ

115 117 119 121 123

61 Please cite this article as: N.H. Nguyen, K. Doğançay, Single-platform passive emitter localization with bearing and Doppler-shift measurements using pseudolinear estimation techniques, Signal Processing (2016), http://dx.doi.org/ 10.1016/j.sigpro.2016.01.023i

N.H. Nguyen, K. Doğançay / Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

12

1

vp;k ¼ vp;1

3

where "

5

Mk ¼

7

11

Here t k ¼ ðk  1ÞΔT where ΔT is the time step. The objective of the DB-TMA problem is to estimate the 4  1 target motion parameter vector ξ ¼ ½pT1 ; vTp;1 T , i.e., the vector of initial target position and velocity. The sensor measurements at time instant k are given by

13

~ k ¼ Θk þ nΘ;k Θ

ð76aÞ

15

Z~ k ¼ Z k þ nZ;k

ð76bÞ

17

where

9

19 21

1 0

ð74bÞ

0 1

23

0 : tk

ð75Þ

pk ð2Þ rk ð2Þ pk ð1Þ rk ð1Þ

ð77aÞ

ðpk  rk ÞT ðvp;k vr;k Þ : J pk  rk J

ð77bÞ

Θk ¼ tan  1

Zk ¼

tk 0

#

29

Note that, to distinguish the notation for the bearing and Doppler-shift measurements of the DB-TMA problem, the upper-case symbols Θ (Theta) and Z (Zeta) are used in this Appendix instead of the lower-case symbols θ (theta) and ζ (zeta) as in the main body of the paper. Writing the sensor measurements in vector form yields

31

Ψ~ ¼ Ψ þ nΨ ¼ ½Θ1 ; Z 1 ; …; ΘN ; Z N T þ ½nΘ;1 ; nZ;1 ; …; nΘ;N ; nZ;N T

33

with the error covariance matrix

35

Σ ¼ EfnΨ nTΨ g ¼ diagðσ 2Θ;1 ; σ 2Z;1 ; …; σ 2Θ;N ; σ 2Z;N Þ:

37

Here nΘ;k and nZ;k are assumed to be zero-mean independent Gaussian random variables with the variances of σ 2Θ;k and σ 2Z;k , respectively. The CRLB for the DB-TMA problem is given by

25 27

39

ð78Þ

ð79Þ

41

CRLBðTMAÞ ¼ ðJTo Σ  1 Jo Þ  1

43

where Jo is the Jacobian matrix of Ψ with respect to ξ evaluated at the true value of ξ:

45

Jo ¼ ½JTΘ;1 ; JTZ;1 ; …; JTΘ;N ; JTZ;N T

47

and

49

JΘ;k ¼

½  sin Θk ; cos Θk Mk J pk rk J

51 53 55 57 59 61

ð80Þ

JZ;k ¼ ½0; 0; cos Θk ; sin Θk  þ

ð81Þ

ð82aÞ ½ak ; bk Mk : J pk  rk J

ð82bÞ

The expressions of ak and bk in (82b) are given by ak ¼ ðvp;k ð1Þ  vr;k ð1ÞÞ sin 2 Θk  ðvp;k ð2Þ  vr;k ð2ÞÞ sin Θk cos Θk ð83aÞ bk ¼ ðvp;k ð2Þ  vr;k ð2ÞÞ cos 2 Θk ðvp;k ð1Þ vr;k ð1ÞÞ sin Θk cos Θk : ð83bÞ

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63

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Please cite this article as: N.H. Nguyen, K. Doğançay, Single-platform passive emitter localization with bearing and Doppler-shift measurements using pseudolinear estimation techniques, Signal Processing (2016), http://dx.doi.org/ 10.1016/j.sigpro.2016.01.023i

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