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...main mathematical tools that are used in this Subsection come from algebraic geometry because a(τ ), b(τ ) and c(τ ) are polynomials with real coefficients (see Appendix B for a short introduction or =-=[13,22,29]-=-). Let us first describe the vanishing locus of c(τ ), over both R, where it is particularly simple, and C. Proposition 6.4 c(τ ) ≥ 0 for every τ ∈ R2. Moreover, c(τ ) = 0 if and only if τ ∈ {R0, R∗, ...

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...on of the previous paragraph, it is well known that the estimated τ̄ ∈ Im(τ∗2 ) given by MLE is the orthogonal projection of τ∗ on the noiseless measurements set, i.e. the projection pH(τ∗) on H (see =-=[6,42]-=- and Fig. 12) therefore x̄ = τ∗2 −1(pH(τ∗)). A similar reasoning applies to the more complex case of τ∗n. In particular, any estimator has a geometrical interpretation and the relative accuracy depend...

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...similar for both cases of generic and collinear microphone configurations, which is the reason why we collect them together in this Section. For the definition and properties of convex polytopes, see =-=[39]-=- among the many available references. Geometry of TDOA maps 13 Theorem 5.1 Im(τ2) is locally the τ–plane. Proof. Let us assume that x̄ is a point where τ2 is regular, i.e. where the Jacobian matrix J(...

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... 6.16 can be nicely interpreted in terms of the two-dimensional and the three-dimensional intersection problems. Here we use some standard Minkowski and relativistic conventions used, for example, in =-=[3, 43]-=-. (i) τ ∈ E if, and only if, v(τ ) is isotropic, or light–like. In this case, the line L21(τ ) is parallel to a generatrix of the cone, therefore it meets C0 at an ideal point. On the x–plane this mea...

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...main mathematical tools that are used in this Subsection come from algebraic geometry because a(τ ), b(τ ) and c(τ ) are polynomials with real coefficients (see Appendix B for a short introduction or =-=[13,22,29]-=-). Let us first describe the vanishing locus of c(τ ), over both R, where it is particularly simple, and C. Proposition 6.4 c(τ ) ≥ 0 for every τ ∈ R2. Moreover, c(τ ) = 0 if and only if τ ∈ {R0, R∗, ...

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... a solution to the localization problem [14]. This is why research has focused on techniques that are aimed at overcoming this problem while achieving robustness. Examples are Maximum Likelihood (ML) =-=[16, 26, 53]-=-; Least Squares (LS) [2]; and Constrained Least Squares (CLS) [34, 48], which offer accurate results for the most common configurations of sensors. There are many situations, however, in which it is n...

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... a solution to the localization problem [14]. This is why research has focused on techniques that are aimed at overcoming this problem while achieving robustness. Examples are Maximum Likelihood (ML) =-=[16, 26, 53]-=-; Least Squares (LS) [2]; and Constrained Least Squares (CLS) [34, 48], which offer accurate results for the most common configurations of sensors. There are many situations, however, in which it is n...

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... a solution to the localization problem [14]. This is why research has focused on techniques that are aimed at overcoming this problem while achieving robustness. Examples are Maximum Likelihood (ML) =-=[16, 26, 53]-=-; Least Squares (LS) [2]; and Constrained Least Squares (CLS) [34, 48], which offer accurate results for the most common configurations of sensors. There are many situations, however, in which it is n...

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...). Quite clearly, on the complex Geometry of TDOA maps 23 plane Ẽ is bound to have singular points, as Ẽ is an algebraic rational quintic curve. In Appendix C we include the source code in Singular =-=[23]-=- language for computing the Cartesian equation (further analysis of the properties of the bifurcation curve is contained in [21]). From Fig. 9 and Fig. 10, we immediately recognize what was assessed t...

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...used on techniques that are aimed at overcoming this problem while achieving robustness. Examples are Maximum Likelihood (ML) [16, 26, 53]; Least Squares (LS) [2]; and Constrained Least Squares (CLS) =-=[34, 48]-=-, which offer accurate results for the most common configurations of sensors. There are many situations, however, in which it is necessary to minimize the number of sensors in use, due to specific sen...

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...lization of a radio-beacon receiver in LORAN navigation systems [49] and in the context of the Global Positioning System (GPS), where measurements are of Time Of Arrivals (TOAs) instead of TDOAs (see =-=[1,8,9,15,18,19,28,30,37,38]-=-). In particular, these studies provide the solution for the case of planar (2D) source localization with three receivers (i.e. with two TDOAs) and they recognize the possibility of dual solutions in ...

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... expressions for the boundaries of the regions of interest. We observe that this approach to the problem fits into the research field of structural identifiability of complex systems (see for example =-=[11, 40]-=-), where one is interested in studying if the parameters of a model (in our case, the coordinates of the source) can be fully retrieved from the experimental data. A similar analysis of the source loc...

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...d how many there are) as the measurements or the sensor geometry vary. This problem has been partially addressed in the case of the localization of a radio-beacon receiver in LORAN navigation systems =-=[49]-=- and in the context of the Global Positioning System (GPS), where measurements are of Time Of Arrivals (TOAs) instead of TDOAs (see [1,8,9,15,18,19,28,30,37,38]). In particular, these studies provide ...

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...lacement. This prospective investigation, in fact, is in line with the recently revamped interest of the research community in self-calibrating and self-synchronizing spatial distributions of sensors =-=[16,45,47]-=-. Our analysis starts from [20], where a different perspective on the localization problem is offered through the adoption of the Space–Range Differences (SRD) reference frame, where the wavefront pro...

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...ents, and τ2 : Ũ0 → U0 is a cover. Let us now assume that the two inverse images of τ0 ∈ U0 belong to the same connected component of Ũ0. As U0 is path–connected, from the Path Lifting Theorem (see =-=[36]-=-), it follows that the inverse images of any other point τ ∈ U0 belong to the same connected component of Ũ0 as well. Let x ′ be a point in the other connected component of Ũ0, with τ ′ = τ2(x′). He...

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...main mathematical tools that are used in this Subsection come from algebraic geometry because a(τ ), b(τ ) and c(τ ) are polynomials with real coefficients (see Appendix B for a short introduction or =-=[13,22,29]-=-). Let us first describe the vanishing locus of c(τ ), over both R, where it is particularly simple, and C. Proposition 6.4 c(τ ) ≥ 0 for every τ ∈ R2. Moreover, c(τ ) = 0 if and only if τ ∈ {R0, R∗, ...

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...roblem [14]. This is why research has focused on techniques that are aimed at overcoming this problem while achieving robustness. Examples are Maximum Likelihood (ML) [16, 26, 53]; Least Squares (LS) =-=[2]-=-; and Constrained Least Squares (CLS) [34, 48], which offer accurate results for the most common configurations of sensors. There are many situations, however, in which it is necessary to minimize the...

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... expressions for the boundaries of the regions of interest. We observe that this approach to the problem fits into the research field of structural identifiability of complex systems (see for example =-=[11, 40]-=-), where one is interested in studying if the parameters of a model (in our case, the coordinates of the source) can be fully retrieved from the experimental data. A similar analysis of the source loc...

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...lization of a radio-beacon receiver in LORAN navigation systems [49] and in the context of the Global Positioning System (GPS), where measurements are of Time Of Arrivals (TOAs) instead of TDOAs (see =-=[1,8,9,15,18,19,28,30,37,38]-=-). In particular, these studies provide the solution for the case of planar (2D) source localization with three receivers (i.e. with two TDOAs) and they recognize the possibility of dual solutions in ...

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...ave been developed and applied to to similar situations, in cases where scientific and engineering models are expressed as sets of real solutions to systems of polynomial equations (see, for example, =-=[4, 25, 35]-=-). The somewhat surprising fact that, although the TDOA map is not polynomial, all the loci involved in the analysis of τ∗2 are algebraic or semi–algebraic, is a promising indicator of the effectivene...

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...lacement. This prospective investigation, in fact, is in line with the recently revamped interest of the research community in self-calibrating and self-synchronizing spatial distributions of sensors =-=[16,45,47]-=-. Our analysis starts from [20], where a different perspective on the localization problem is offered through the adoption of the Space–Range Differences (SRD) reference frame, where the wavefront pro...

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...τ∗2 . The complete model τ̂∗2 (x) takes into account each one of the three TDOA that can be defined between the sensors. This, in fact, becomes necessary when working in a realistic (noisy) situation =-=[50]-=-. We should keep in mind, however, that there is a linear relationship between the pseudoranges (3), which allows us to simplify the deterministic problem. Definition 2.2 Let (τ10, τ20, τ21) be the co...

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...lization of a radio-beacon receiver in LORAN navigation systems [49] and in the context of the Global Positioning System (GPS), where measurements are of Time Of Arrivals (TOAs) instead of TDOAs (see =-=[1,8,9,15,18,19,28,30,37,38]-=-). In particular, these studies provide the solution for the case of planar (2D) source localization with three receivers (i.e. with two TDOAs) and they recognize the possibility of dual solutions in ...

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...ave been developed and applied to to similar situations, in cases where scientific and engineering models are expressed as sets of real solutions to systems of polynomial equations (see, for example, =-=[4, 25, 35]-=-). The somewhat surprising fact that, although the TDOA map is not polynomial, all the loci involved in the analysis of τ∗2 are algebraic or semi–algebraic, is a promising indicator of the effectivene...

Citation Context

...lization of a radio-beacon receiver in LORAN navigation systems [49] and in the context of the Global Positioning System (GPS), where measurements are of Time Of Arrivals (TOAs) instead of TDOAs (see =-=[1,8,9,15,18,19,28,30,37,38]-=-). In particular, these studies provide the solution for the case of planar (2D) source localization with three receivers (i.e. with two TDOAs) and they recognize the possibility of dual solutions in ...

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...lization of a radio-beacon receiver in LORAN navigation systems [49] and in the context of the Global Positioning System (GPS), where measurements are of Time Of Arrivals (TOAs) instead of TDOAs (see =-=[1,8,9,15,18,19,28,30,37,38]-=-). In particular, these studies provide the solution for the case of planar (2D) source localization with three receivers (i.e. with two TDOAs) and they recognize the possibility of dual solutions in ...

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... 6.16 can be nicely interpreted in terms of the two-dimensional and the three-dimensional intersection problems. Here we use some standard Minkowski and relativistic conventions used, for example, in =-=[3, 43]-=-. (i) τ ∈ E if, and only if, v(τ ) is isotropic, or light–like. In this case, the line L21(τ ) is parallel to a generatrix of the cone, therefore it meets C0 at an ideal point. On the x–plane this mea...

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... case, the coordinates of the source) can be fully retrieved from the experimental data. A similar analysis of the source localization problem has been proposed and investigated very recently also in =-=[5, 17]-=-, the latter in the context of the TOA–based target tracking. We believe that characterizing the TDOA map to its fullest extent, even in the simplest case of three calibrated and synchronous sensors, ...

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...e localization but measurement errors cause the corresponding hyperbola branches not to meet at a single point, thus ruling out simple geometric intersection as a solution to the localization problem =-=[14]-=-. This is why research has focused on techniques that are aimed at overcoming this problem while achieving robustness. Examples are Maximum Likelihood (ML) [16, 26, 53]; Least Squares (LS) [2]; and Co...

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...lization of a radio-beacon receiver in LORAN navigation systems [49] and in the context of the Global Positioning System (GPS), where measurements are of Time Of Arrivals (TOAs) instead of TDOAs (see =-=[1,8,9,15,18,19,28,30,37,38]-=-). In particular, these studies provide the solution for the case of planar (2D) source localization with three receivers (i.e. with two TDOAs) and they recognize the possibility of dual solutions in ...

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... case, the coordinates of the source) can be fully retrieved from the experimental data. A similar analysis of the source localization problem has been proposed and investigated very recently also in =-=[5, 17]-=-, the latter in the context of the TOA–based target tracking. We believe that characterizing the TDOA map to its fullest extent, even in the simplest case of three calibrated and synchronous sensors, ...

Citation Context

...lization of a radio-beacon receiver in LORAN navigation systems [49] and in the context of the Global Positioning System (GPS), where measurements are of Time Of Arrivals (TOAs) instead of TDOAs (see =-=[1,8,9,15,18,19,28,30,37,38]-=-). In particular, these studies provide the solution for the case of planar (2D) source localization with three receivers (i.e. with two TDOAs) and they recognize the possibility of dual solutions in ...

Citation Context

...lization of a radio-beacon receiver in LORAN navigation systems [49] and in the context of the Global Positioning System (GPS), where measurements are of Time Of Arrivals (TOAs) instead of TDOAs (see =-=[1,8,9,15,18,19,28,30,37,38]-=-). In particular, these studies provide the solution for the case of planar (2D) source localization with three receivers (i.e. with two TDOAs) and they recognize the possibility of dual solutions in ...

Citation Context

...ion, in fact, is in line with the recently revamped interest of the research community in self-calibrating and self-synchronizing spatial distributions of sensors [16,45,47]. Our analysis starts from =-=[20]-=-, where a different perspective on the localization problem is offered through the adoption of the Space–Range Differences (SRD) reference frame, where the wavefront propagation is described by a (pro...

Citation Context

...on of the previous paragraph, it is well known that the estimated τ̄ ∈ Im(τ∗2 ) given by MLE is the orthogonal projection of τ∗ on the noiseless measurements set, i.e. the projection pH(τ∗) on H (see =-=[6,42]-=- and Fig. 12) therefore x̄ = τ∗2 −1(pH(τ∗)). A similar reasoning applies to the more complex case of τ∗n. In particular, any estimator has a geometrical interpretation and the relative accuracy depend...

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...rs (i.e. with two TDOAs) and they recognize the possibility of dual solutions in some instances, as two different source positions could correspond to the same pair of TDOA measurements. Recently, in =-=[51]-=- the author focused on the assessment of the ill-posedness of the localization problem in the case of 2D minimal sensor configurations, i.e. on quantify how changes in the measurements propagate onto ...

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...ie on the surface of the propagation cone, localizing a source in the SRD space corresponds to finding the vertex of the cone that best fits the measured data. The SRD reference frame is also used in =-=[12]-=- to offer geometric interpretations to the underlying principles behind the most common TDOA-based localization solutions. Although not explicitly claimed, the localization problem is described in [12...

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...nal quintic curve. In Appendix C we include the source code in Singular [23] language for computing the Cartesian equation (further analysis of the properties of the bifurcation curve is contained in =-=[21]-=-). From Fig. 9 and Fig. 10, we immediately recognize what was assessed through simulations and for a specific sensor configuration in [51]. These results, however, have been here derived in closed for...

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...used on techniques that are aimed at overcoming this problem while achieving robustness. Examples are Maximum Likelihood (ML) [16, 26, 53]; Least Squares (LS) [2]; and Constrained Least Squares (CLS) =-=[34, 48]-=-, which offer accurate results for the most common configurations of sensors. There are many situations, however, in which it is necessary to minimize the number of sensors in use, due to specific sen...

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...In particular, any estimator has a geometrical interpretation and the relative accuracy depends on the (non trivial) shape of Im(τ∗n). Possible techniques to be applied come from Information Geometry =-=[6, 35]-=-, an approach that proved successful in similar situations and that is based on the careful description of Im(τ∗n). With this in mind, Geometry of TDOA maps 36 notice that our characterization of the ...

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...lization of a radio-beacon receiver in LORAN navigation systems [49] and in the context of the Global Positioning System (GPS), where measurements are of Time Of Arrivals (TOAs) instead of TDOAs (see =-=[1,8,9,15,18,19,28,30,37,38]-=-). In particular, these studies provide the solution for the case of planar (2D) source localization with three receivers (i.e. with two TDOAs) and they recognize the possibility of dual solutions in ...