Bayesian FDOA-Only Localization Under Correlated Measurement Noise: A Low-Complexity Gaussian Conditional-Based Approach

Abstract

This paper presents the Gaussian conditional method (GCM) for the problem of frequency difference of arrival (FDOA)-only source localization under correlated noise. GCM identifies the source position through approximating its posterior distribution using a Gaussian mixture model (GMM) and applying successive conditioning to the measurement likelihood. The algorithm development leverages the fact that FDOA measurements follow a multivariate Gaussian distribution with a non-diagonal covariance. Simulation results demonstrate that GCM can achieve the Cramér–Rao lower bound (CRLB) under moderate noise levels, while having lower computational complexity than baseline techniques including the recently developed Gaussian division method (GDM). The proposed algorithm is particularly effective for passively locating narrowband sources, where the time difference of arrival (TDOA) measurements become unreliable, and it can operate without the need for accurate initialization.

1. Introduction

Passive localization enables estimating the positions of sources without active illuminations, making it particularly attractive for covert and resource-constrained scenarios [1,2,3,4]. Commonly employed measurements include the time of arrival (TOA) [5], angle of arrival (AOA) [6], received signal strength (RSS) [7], TDOA [8,9], phase difference of arrival (PDOA) [10,11], and FDOA [12]. These measurements can be used individually or jointly to determine the source location. Among them, FDOA measurements are particularly effective for locating narrowband sources where TDOA-based approaches become unreliable [13,14]. This motivates the study of the FDOA-only localization problem, which despite its practical importance, remains challenging due to the strong nonlinearity of the FDOA measurement equations and correlated measurement noise [15,16].

Among existing studies, grid-search (GS) methods [17] suffer from rapidly increasing computational complexity as accuracy requirements grow, while Gauss Newton (GN) iterative approaches [18] rely heavily on precise initialization and graph-based methods [19,20] fail to meet real-time demands. The FDOA-random-sample-consensus (FDOAR) algorithm [21] improves robustness against outliers by selectively using subsets of measurements, but this comes at the cost of reduced accuracy since part of the available information is discarded. More recently, two-step weighted least squares (TSWLS) solutions [22,23] have been proposed to overcome the lack of closed-form formulations for FDOA-only localization. However, these methods introduce multiple intermediate variables and require an excessive number of observers, at least 14 in two-dimensional and 20 in three-dimensional scenarios, thereby limiting their practical applicability.

In our previous work [24], GDM was proposed as a new Bayesian solution for FDOA-only localization. However, the method faces limitations due to the auxiliary covariance matrix construction and computational complexity. To overcome these limitations, we propose the GCM, an enhanced Bayesian estimation approach that directly addresses the correlation among FDOA measurements. By applying the FDOA measurements to refine the source position posterior in a successive manner, GCM eliminates the need for constructing an auxiliary noise covariance matrix, as required by GDM. Specifically, the measurement likelihood is the Gaussian conditional of an FDOA, given all the previously applied FDOAs for updating the posterior. This iterative process leads to reduced computational complexity. Numerical simulations confirm that GCM is capable of achieving CRLB accuracy under moderate noise levels, making it a more efficient choice compared to existing methods for FDOA-only localization.

The remainder of this paper is organized as follows. Section 2 formulates the localization problem. Section 3 presents the proposed GCM algorithm. Section 4 analyzes the computational complexity. Section 5 provides numerical simulation results. Finally, Section 6 concludes the paper.

2. Problem Formulation

As illustrated in Figure 1, we consider a two-dimensional FDOA-only localization scenario similar to that in [24]. A total of $N+1$ moving observers are used to locate a stationary source at $\mathbf{u}={[x_t,y_t]}^T$, where ${(·)}^T$ denotes the transpose operation. The source transmits a signal with carrier frequency $f_c$. Let ${\mathbf{s}}_i$ and ${\dot{\mathbf{s}}}_i$ denote the position and velocity of the i-th observer $(i=0,1,\dots ,N)$, and observer 0 is chosen as the reference. The FDOA measurement between observer i and the reference is modeled as

(1a)$\begin{array}{c}{z}_i=h_i\left(\mathbf{u}\right)+n_i,\end{array}$

(1b)$\begin{array}{c}{h}_i\left(\mathbf{u}\right)={\displaystyle \frac{f_c}{c}}\left({\displaystyle \frac{{(\mathbf{u}-{\mathbf{s}}_i)}^T{\dot{\mathbf{s}}}_i}{\parallel \mathbf{u}-{\mathbf{s}}_i\parallel }}-{\displaystyle \frac{{(\mathbf{u}-{\mathbf{s}}_0)}^T{\dot{\mathbf{s}}}_0}{\parallel \mathbf{u}-{\mathbf{s}}_0\parallel }}\right),\end{array}$

where $i=1,\dots ,N$, c is the signal propagation speed and $n_i$ denotes zero-mean Gaussian noise.

Stacking all the N FDOA measurements gives

(2)$\mathbf{z}=\mathbf{h}\left(\mathbf{u}\right)+\mathbf{n},$

with noise covariance

(3)$\mathbf{R}=\mathbb{E}\left[\mathbf{n}{\mathbf{n}}^T\right]={\sigma }_f^2\mathbf{Q}.$

Here, $\mathbb{E}[·]$ denotes the expectation operator, ${\sigma }_f^2$ is the variance of the measurement noise, and $\mathbf{Q}$ represents the covariance structure matrix. The off-diagonal elements of $\mathbf{Q}$ characterize the correlation among the measurement noises. As an example, the covariance structure matrix

(4)$\mathbf{Q}=\left[\begin{array}{cccc}1& \beta & \dots & \beta \\ \beta & 1& \dots & \beta \\ ⋮& ⋮& \ddots & ⋮\\ \beta & \beta & \dots & 1\end{array}\right],$

has been adopted in [25] to model the scenario in which all observers have identical configurations, with $\beta \in [0,1)$ denoting the correlation coefficient between different measurement noises.

3. Gaussian Conditional Method

This section details the proposed GCM, an improved Bayesian localization framework that estimates the source position solely on the basis of FDOA measurements. Once we obtain the source position posterior $p\left(\mathbf{u}\right|\mathbf{z})$, the source position estimate may be found through computing the posterior mean via

(5)$\widehat{\mathbf{u}}\approx \int \mathbf{u}p\left(\mathbf{u}\right|\mathbf{z})d\mathbf{u}.$

To compute the desired posterior $p\left(\mathbf{u}\right|\mathbf{z})$, by Bayes’ theorem, it can be expressed as

(6)$\begin{array}{c}\hfill p\left(\mathbf{u}|\mathbf{z}\right)\propto p\left(\mathbf{z}\right|\mathbf{u})p\left(\mathbf{u}\right)\propto p\left({\mathbf{z}}_{2:N}\right|\mathbf{u})p\left(\mathbf{u}\right|z_1)\propto \prod _{i=2}^{N}p\left(z_i\right|\mathbf{u},{\mathbf{z}}_{1:i-1})p\left(\mathbf{u}\right|z_1),\end{array}$

where ${\mathbf{z}}_{1:i-1}={[z_1,\dots ,z_{i-1}]}^T$. It is worthwhile to point out that starting with $z_1$ to find the source position posterior is actually arbitrary.

The measurement likelihood $p\left(z_i\right|\mathbf{u},{\mathbf{z}}_{1:i-1})$ is, by Gaussian conditional [26],

(7)$\begin{array}{cc}\hfill & p\left(z_i\right|\mathbf{u},{\mathbf{z}}_{1:i-1})={\displaystyle \frac{p\left({\mathbf{z}}_{1:i}\right|\mathbf{u})}{p\left({\mathbf{z}}_{1:i-1}\right|\mathbf{u})}}\hfill \\ \hfill & ={\displaystyle \frac{\mathcal{N}\left(\left[\begin{array}{c}{\mathbf{z}}_{1:i-1}\\ z_i\end{array}\right];\left[\begin{array}{c}{\mathbf{h}}_{1:i-1}\left(\mathbf{u}\right)\\ h_i\left(\mathbf{u}\right)\end{array}\right],\left[\begin{array}{cc}{\mathbf{R}}_{i-1}& {\mathbf{c}}_{i-1}\\ {\mathbf{c}}_{i-1}^T& {\sigma }_i^2\end{array}\right]\right)}{\mathcal{N}({\mathbf{z}}_{1:i-1};{\mathbf{h}}_{1:i-1}\left(\mathbf{u}\right),{\mathbf{R}}_{i-1})}}\hfill \\ \hfill & =\mathcal{N}(z_i;{\widehat{z}}_i,{\widehat{\sigma }}_i^2),\hfill \end{array}$

with ${\mathbf{R}}_{i-1}=\mathbb{E}\left({\mathbf{n}}_{1:i-1}{\mathbf{n}}_{1:i-1}^T\right)$, ${\mathbf{c}}_{i-1}=\mathbb{E}\left({\mathbf{n}}_{1:i-1}{n}_i\right)$, ${\sigma }_i^2=\mathbb{E}\left(n_i^2\right)$ and ${\mathbf{n}}_{1:i-1}={[n_1,n_2,...,n_{i-1}]}^T$. In addition, we have

(8a)$\begin{array}{c}{\widehat{z}}_i=h_i\left(\mathbf{u}\right)+{\mathbf{c}}_{i-1}^T{\mathbf{R}}_{i-1}^{-1}({\mathbf{z}}_{1:i-1}-{\mathbf{h}}_{1:i-1}\left(\mathbf{u}\right)),\end{array}$

(8b)$\begin{array}{c}{\widehat{\sigma }}_i^2={\sigma }_i^2-{\mathbf{c}}_{1:i-1}^T{\mathbf{R}}_{i-1}^{-1}{\mathbf{c}}_{i-1},\end{array}$

where ${\mathbf{h}}_{1:i-1}\left(\mathbf{u}\right)={[h_1\left(\mathbf{u}\right),h_2\left(\mathbf{u}\right),...,h_{i-1}\left(\mathbf{u}\right)]}^T$.

From (8a), we can observe that ${\widehat{z}}_i$ remains a nonlinear function of the unknown source position $\mathbf{u}$, as expected. But it also depends on other FDOAs, reflecting how the correlation among the measurement noise is taken into consideration. Both (8a) and (8b) require inverting of the covariance matrix ${\mathbf{R}}_{i-1}$. This can be done in advance or through exploiting the partitioned form of ${\mathbf{R}}_{i-1}$ [27], which is

(9)${\mathbf{R}}_{i-1}=\left[\begin{array}{cc}{\mathbf{R}}_{i-2}& {\mathbf{c}}_{i-2}\\ {\mathbf{c}}_{i-2}^T& {\sigma }_{i-1}^2\end{array}\right].$

Applying the Schur complement and defining ${\mathbf{q}}_{i-2}\triangleq {\mathbf{R}}_{i-2}^{-1}{\mathbf{c}}_{i-2}$, we arrive at

(10)${\mathbf{R}}_{i-1}^{-1}=\left[\begin{array}{cc}{\mathbf{R}}_{i-2}^{-1}+\frac{1}{{\widehat{\sigma }}_{i-1}^2}{\mathbf{q}}_{i-2}{\mathbf{q}}_{i-2}^T& -\frac{1}{{\widehat{\sigma }}_{i-1}^2}{\mathbf{q}}_{i-2}\\ -\frac{1}{{\widehat{\sigma }}_{i-1}^2}{\mathbf{q}}_{i-2}^T& \frac{1}{{\widehat{\sigma }}_{i-1}^2}\end{array}\right],$

where

(11)${\widehat{\sigma }}_{i-1}^2={\sigma }_{i-1}^2-{\mathbf{c}}_{i-2}^T{\mathbf{q}}_{i-2}.$

This recursion starts with $i=2$ and ${\mathbf{R}}_1^{-1}={\displaystyle \frac{1}{{\sigma }_1^2}}$.

The required initial posterior $p\left(\mathbf{u}\right|z_1)$ in (6) is obtained using the same method as in [24], which is expressed in terms of a GMM [28] given by

(12)$p\left(\mathbf{u}\right|z_1)=\sum _{g=1}^{N_g}{\widehat{\omega }}_{1|1}^g\mathcal{N}(\mathbf{u};{\widehat{\mathbf{u}}}_{1|1}^g,{\widehat{\mathbf{P}}}_{1|1}^g).$

Here, $N_g$ denotes the number of Gaussian components. The weights satisfy ${\widehat{\omega }}_{1|1}^g>0$ and ${\sum }_{g=1}^{N_g}{\widehat{\omega }}_{1|1}^g=1$.

Substituting (7) and (12) into (6) yields

(13)$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}p\left(\mathbf{u}\right|\mathbf{z})& \propto \sum _{g=1}^{N_g}\prod _{i=2}^N\mathcal{N}(z_i;{\widehat{z}}_i,{\widehat{\sigma }}_i^2){\widehat{\omega }}_{1|1}^g\mathcal{N}(\mathbf{u};{\widehat{\mathbf{u}}}_{1|1}^g,{\widehat{\mathbf{P}}}_{1|1}^g)\hfill \\ \hfill & =\sum _{g=1}^{N_g}{\omega }_{N|N}^g\mathcal{N}(\mathbf{u};{\widehat{\mathbf{u}}}_{N|N}^g,{\mathbf{P}}_{N|N}^g),\hfill \end{array}$

where the equality follows from invoking the measurement update stage of, e.g., cubature Kalman filter (CKF) [29] $N-1$ times successively. The weights ${\omega }_{N|N}^g$ are positive and normalized such that ${\sum }_{g=1}^{N_g}{\omega }_{N|N}^g=1$. They are in fact the marginal measurement likelihood, which can also be computed using numerical integration techniques such as the one based on the spherical–radial cubature rule and adopted by the CKF. The specific CKF update process can be found in [24], and will not be repeated here.

It is worth noting that the proposed GCM naturally fits within the general Bayesian filtering framework. In a standard Bayesian filter, the posterior distribution is recursively updated as

(14)$p\left(\mathbf{u}\right|{\mathbf{z}}_{1:i})\propto p\left(z_i\right|\mathbf{u},{\mathbf{z}}_{1:i-1})p\left(\mathbf{u}\right|{\mathbf{z}}_{1:i-1}),$

where $p\left(\mathbf{u}\right|{\mathbf{z}}_{1:i-1})$ is the prior at step i, and $p\left(z_i\right|\mathbf{u},{\mathbf{z}}_{1:i-1})$ denotes the measurement likelihood. Equation (6) directly follows from (14), with the likelihood explicitly derived through Gaussian conditional (see (7) and (8)). Unlike conventional Kalman-type filters assuming independent measurement noise, GCM incorporates correlation by conditioning each FDOA on previous FDOA measurements, enabling sequential updates without reconstructing the full joint covariance. When the likelihoods are approximated using the spherical–radial cubature rule, the update step of GCM becomes equivalent to that of the CKF [29], while the recursive form in (10) further reduces matrix inversion complexity. Thus, GCM adheres to the Bayesian inference principle while maintaining high computational efficiency for correlated FDOA measurements.

4. Computational Complexity Analysis

The proposed GCM algorithm consists of two main stages: the initialization stage, which is identical to that in [24], and the iterative CKF-based update stage. As discussed in [24], the initialization stage constructs $N_g$ Gaussian components, leading to a computational complexity of $\mathcal{O}\left(N_g\right)$. During each CKF update iteration, when processing the i-th FDOA measurement, the inverse of the covariance matrix ${\mathbf{R}}_{i-1}^{-1}$ needs to be constructed. At this step, the upper-left submatrix of ${\mathbf{R}}_{i-1}^{-1}$ has a dimension of $(i-2)\times (i-2)$. The computation of ${\mathbf{q}}_{i-2}$ involves a matrix–vector multiplication, resulting in a complexity of $\mathcal{O}\left({(i-2)}^2\right)$. The computation of (11) involves a vector inner product, with a complexity of $\mathcal{O}(i-2)$. Furthermore, constructing the upper-left submatrix of (10) requires one outer product and one matrix addition, both of which incur $\mathcal{O}\left({(i-2)}^2\right)$ operations. Therefore, the computational complexity of each recursive update is $\mathcal{O}\left(i^2\right)$.

Note that the case of $i=2$ only involves a scalar inversion; its cost is negligible. Summing over all iterations from $i=3$ to N, the total computational complexity of the recursive inversion process is

(15)$\sum _{i=3}^N\mathcal{O}\left(i^2\right)=\mathcal{O}\left(N^3\right).$

Consequently, the overall computational complexity of the proposed GCM algorithm is jointly determined by the number of Gaussian components $N_g$ and the number of FDOA measurements N, leading to a total complexity of $\mathcal{O}\left(N_g{N}^3\right)$.

5. Simulation Results

This section evaluates the localization performance of the proposed GCM algorithm through extensive Monte Carlo simulations. Table 1 presents the observer configuration with their true positions and velocities. The source is located at $\mathbf{u}={[3,10]}^T$ km and transmits a signal with carrier frequency $f_c=3$ GHz and a propagation speed of $3\times {10}^8$ m/s. The number of Gaussian components is set to $N_g=20$, the same as in [24]. The correlation coefficient is set to $\beta =0.5$ by default and varied within $[0,0.9]$ to analyze its effect on localization performance. To further investigate the effect of the accuracy of the measurement used for initialization, a noise scaling factor $\gamma$ is introduced to adjust its variance. Without loss of generality, the first FDOA measurement is used for Gaussian initialization, whose noise variance is set to ${\sigma }_1^2=\gamma {\sigma }_1^2$. The default value of $\gamma$ is 1, and its effect is analyzed by varying $\gamma$ within $[1,2]$.

Performance comparisons are conducted against the GDM algorithm [24], the GN iterative algorithm [18], the GS algorithm [17], and the FDOAR algorithm [21] under varying noise levels. The first stage of the GDM algorithm is denoted as “KF-1”, and the scaling factor in GDM is set as $\alpha =2$ for performance evaluation. For the GN method, the initial estimates are obtained by perturbing the true source position with Gaussian noise whose covariance is set to $10\times \mathrm{CRLB}\left(\mathbf{u}\right)$. For the GS method, the search region is defined as a square centered at the true source position with side length $20\times \mathrm{CRLB}\left(\mathbf{u}\right)$. Two grid resolutions are considered: $2\times \mathrm{CRLB}\left(\mathbf{u}\right)$, referred to as “GS-1”, and $0.5\times \mathrm{CRLB}\left(\mathbf{u}\right)$, referred to as “GS-2”. Localization accuracy is evaluated using the root mean square error (RMSE) computed over M = 10,000 Monte Carlo trials. The CRLB is included as a theoretical benchmark [30], given by

(16)$\mathrm{CRLB}\left(\mathbf{u}\right)={\left[{\left({\displaystyle \frac{\partial \mathbf{h}\left(\mathbf{u}\right)}{\partial \mathbf{u}}}\right)}^T{\mathbf{R}}^{-1}\left({\displaystyle \frac{\partial \mathbf{h}\left(\mathbf{u}\right)}{\partial \mathbf{u}}}\right)\right]}^{-1},$

with the Jacobian given in [24]. All simulations were carried out on an Intel Core i9-13900H processor with 32 GB of RAM using MATLAB R2020b.

The first four observers listed in Table 1 are employed to evaluate the localization performance of different algorithms under varying noise levels. Figure 2 illustrates the localization accuracy of the considered methods, while Table 2 summarizes their average computation times. Both GDM and GCM achieve the CRLB under low-noise conditions, clearly outperforming GN, FDOAR, and GS-1. The GN method approaches the CRLB at low noise levels but deteriorates rapidly as the noise increases, reflecting its sensitivity to initialization. The FDOAR method consistently falls short of the CRLB because, in overdetermined scenarios, it does not fully exploit all available measurements, which is consistent with the observations in [24]. Although GS-2 achieves relatively high localization accuracy, Table 2 shows that its computational cost is substantially higher than that of other algorithms. Under the present simulation conditions, GCM and GDM provide nearly identical localization performance, indicating that both methods yield consistent posterior distributions $p\left(\mathbf{u}\right|\mathbf{z})$ through different estimation strategies. Moreover, GCM achieves an even lower computational cost than KF-1 in the current simulation setting.

Further comparisons are shown in Figure 3, where the localization accuracy of GCM and GDM is evaluated with different numbers of observers. As before, the two methods deliver nearly identical performance, with GCM demonstrating robustness regardless of the number of observers. For 2D localization, GCM achieves accurate results with as few as three observers. Table 3 summarizes the average computational time of these two methods as the number of observers increases. For small numbers of observers, GCM achieves lower computational complexity than KF-1. As the number of observers increases, its complexity surpasses that of KF-1 due to iterative matrix inversions, while still remaining consistently lower than that of GDM.

To evaluate the robustness of the proposed algorithms, the effects of the correlation coefficient $\beta$ and the measurement noise scaling factor $\gamma$ are examined. As illustrated in Figure 4, the localization accuracy of both methods improves as the correlation coefficient $\beta$ increases, indicating that they can effectively adapt to the correlation of measurement noise. From Figure 5, it can be observed that as the measurement noise scaling factor $\gamma$ increases, the localization accuracy of GCM and GDM starts to diverge. When the number of observers is $N_s=3$, both GCM and GDM exhibit reduced accuracy with increasing $\gamma$, with the performance of GDM gradually falling behind that of GCM. However, when the number of observers satisfies $N_s\ge 5$, GDM achieves higher localization accuracy than GCM.

This phenomenon can be attributed to the differences in the localization mechanisms of GDM and GCM. In [24], the auxiliary covariance matrix $\mathbf{D}$ is constructed by inflating the maximum eigenvalue of the original covariance matrix $\mathbf{R}$ under the assumption of independent measurement noise. Although this approach ensures positive definiteness, it simultaneously increases the effective noise level. When the number of observers is small, the subsequent refinement steps cannot effectively mitigate the degradation caused by the amplified noise. As the number of observers increases, this loss is alleviated since the GDM refinement step simultaneously utilizes all available measurements. In contrast, GCM performs sequential updates directly on correlated measurements. Due to the strong nonlinearity of the FDOA equations, GCM implements an approximate filtering process that leads to error accumulation over iterations. With Gaussian conditioning, the FDOA measurements have conditional variances smaller than their original values (see (8b)), causing GCM with poorer initialization to suffer more from approximation errors due to Gaussian filtering. Even with more measurements, error accumulation cannot be effectively mitigated, ultimately leading to lower localization accuracy compared to GDM.

6. Conclusions

This work revisits the problem of FDOA-only localization from a perspective different from that in our previous study [24]. The proposed GCM directly exploits the conditional structure of multivariate Gaussian distributions to perform sequential Bayesian updates. Simulation results demonstrate that GCM attains the CRLB under low-noise conditions and achieves localization accuracy comparable to GDM. In a 2D scenario, it requires as few as three observers. When the initial posterior uncertainty is large, GCM exhibits higher accuracy with a small number of observers but becomes slightly less accurate than GDM as the number of observers increases, due to error accumulation inherent in iterative filtering. GCM also demonstrates strong adaptability to different noise correlation levels. Moreover, it incurs lower computational cost than KF-1 with few observers and maintains moderate complexity between KF-1 and GDM as the observer count increases. Overall, GCM provides a complementary approach that balances localization accuracy with computational efficiency, making it particularly suitable for narrowband scenarios where TDOA information is unavailable or unreliable.

Author Contributions

Conceptualization, X.L. and L.Y.; methodology, W.Z. and X.L.; software, W.Z., Y.L. and X.L.; validation, W.Z., Y.L. and X.L.; formal analysis, W.Z. and Y.L.; writing—original draft, W.Z.; writing—review and editing, X.L., L.Y. and F.G.; supervision, L.Y. and F.G. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data presented in this study are available in this article.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1 Illustration of the FDOA only localization scenario with $N+1$ moving observers. The iso-FDOA curve represents the locus of points where the FDOA measurements are identical.

Figure 2 Localization accuracy of different algorithms under varying measurement noise levels, using the first four observers from Table 1.

Figure 3 Localization accuracy of GDM and GCM under varying numbers of observers ($N_s=3,5,7,9$).

Figure 4 Effect of correlation coefficient $\beta$ on localization accuracy of GDM and GCM for varying numbers of observers (${\sigma }_f=8$ Hz).

Figure 5 Effect of the measurement noise scaling factor $\gamma$ on localization accuracy of GDM and GCM for varying numbers of observers (${\sigma }_f=8$ Hz).

Table 1

Positions and velocities of the observers.

Observer No.	Position (m)	Velocity (m/s)
1	${\mathbf{s}}_1={[0,0]}^T$	${\dot{\mathbf{s}}}_1={[100,0]}^T$
2	${\mathbf{s}}_2={[2000,0]}^T$	${\dot{\mathbf{s}}}_2={[110,4]}^T$
3	${\mathbf{s}}_3={[1500,1000]}^T$	${\dot{\mathbf{s}}}_3={[138,24]}^T$
4	${\mathbf{s}}_4={[-500,500]}^T$	${\dot{\mathbf{s}}}_4={[75,27]}^T$
5	${\mathbf{s}}_5={[3000,3000]}^T$	${\dot{\mathbf{s}}}_5={[142,-5]}^T$
6	${\mathbf{s}}_6={[500,1500]}^T$	${\dot{\mathbf{s}}}_6={[76,-39]}^T$
7	${\mathbf{s}}_7={[-500,4000]}^T$	${\dot{\mathbf{s}}}_7={[92,43]}^T$
8	${\mathbf{s}}_8={[1500,2500]}^T$	${\dot{\mathbf{s}}}_8={[82,9]}^T$
9	${\mathbf{s}}_9={[1000,2500]}^T$	${\dot{\mathbf{s}}}_9={[117,-27]}^T$

Table 2

Average computation times of different algorithms for the four-observer configuration.

Method	Average Time (ms)
GCM	1.70
GDM	2.21
KF-1 (uncorrelated)	1.82
FDOAR	2.73
GN	0.17
GS-1	0.22
GS-2	5.01

Table 3

Average computation times of KF-1, GDM, and GCM under varying numbers of observers.

Number of Observers	KF-1 (ms)	GDM (ms)	GCM (ms)
3	1.45	1.88	1.34
5	2.03	2.51	2.12
7	2.76	3.32	2.97
9	3.47	4.08	3.82

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