TDOA–based Localization in Two Dimensions : the Bifurcation Curve

• Autorzy: Compagnoni M. , Notari R.

Identyfikatory

DOI

10.3233/FI-2014-1118

• Języki publikacji: EN

Abstrakty

EN

In this paper, we complete the study of the geometry of the TDOA map that encodes the noiseless model for the localization of a source from the range differences between three receivers in a plane, by computing the Cartesian equation of the bifurcation curve in terms of the positions of the receivers. From that equation, we can compute its real asymptotic lines. The present manuscript completes the analysis of [12]. Our result is useful to check if a source belongs or is closed to the bifurcation curve, where the localization in a noisy scenario is ambiguous.

Słowa kluczowe

EN

indoor positioning system; bifurcation theory; two-dimensional model; encoding; cartesian plane; asymptote

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2014
• Tom: Vol. 135, nr 1/2
• Strony: 199--210
• Opis fizyczny: Bibliogr. 31 poz., wykr.

Twórcy

autor

Compagnoni M.

• Dipartimento di Matematica, Politecnico di Milano, I-20133 Milano, Italia

autor

Notari R.

• Dipartimento di Matematica, Politecnico di Milano, I-20133 Milano, Italia

Bibliografia

• [1] Abel, J., Chauffe, J.: Existence and uniqueness of GPS solutions, IEEE Transactions on Aerospace and Electronic Systems, 27, November 1991, 952–956.
• [2] Abel, J., Smith, J.: The spherical interpolation method for closed-form passive source localization using range difference measurements, Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP ’87., 12, apr 1987.
• [3] Amari, S., Nagaoka, H.: Methods of Information Geometry, American Mathematical Society, 2000.
• [4] Awange, J., Shan, J.: Algebraic Solution of GPS Pseudo-Ranging Equations, GPS Solutions, 5(4), 2002, 20-32.
• [5] Bancroft, S.: An Algebraic Solution of the GPS Equations, IEEE Transactions on Aerospace Electronic Systems, 21, January 1985, 56–59.
• [6] Beck, A., Stoica, P., Li, J.: Exact and Approximate Solutions of Source Localization Problems, Signal Processing, IEEE Transactions on, 56(5), May 2008, 1770 –1778, ISSN 1053-587X.
• [7] Bestagini, P., Compagnoni, M., Antonacci, F., Sarti, A., Tubaro, S.: TDOA-Based Acoustic Source Localization in the Space–Range Reference Frame, Multidimensional Systems and Signal Processing, 2013.
• [8] Boyer, C.: A History of Mathematics, New York: Wiley, 1989.
• [9] Chauffe, J., Abel, J.: On the exact solution of the pseudorange equations, IEEE Transactions on Aerospace and Electronic Systems, 30, October 1994, 1021–1030.
• [10] Coll, B., Ferrando, J., Morales-Lladosa, J.: Positioning systems in Minkowski space-time: from emission to inertial coordinates, Classical Quantum Gravity, 27, 2010, 065013.
• [11] Coll, B., Ferrando, J., Morales-Lladosa, J.: Positioning systems in Minkowski space-time: Bifurcation problem and observational data, Phys. Rev. D, 86, Oct 2012, 084036.
• [12] Compagnoni, M., Notari, R., Antonacci, F., Sarti, A.: A comprehensive analysis of the geometry of TDOA maps in localization problems, Inverse Problems, 30(3), 2014, 035004.
• [13] Draisma, J., Horobet, E., Ottaviani, G., Sturmfels, B., Thomas, R.: The Euclidean distance degree of an algebraic variety, 2013.
• [14] Getting, I.: The Global Positioning System, IEEE Spectrum, SPEC-30, December 1993, 36–47.
• [15] Gillette, M., Silverman, H.: A Linear Closed-Form Algorithm for Source Localization From Time-Differences of Arrival, IEEE Signal Processing Letters, 15, 2008, 1–4.
• [16] Grafarend, E., Shan, J.: GPS Solutions: Closed Forms, Critical and Special Configurations of P4P, GPS Solutions, 5(3), 2002, 29–41.
• [17] Hoshen, J.: The GPS Equations and the Problem of Apollonius, IEEE Transactions on Aerospace and Electronic Systems, 32(3), July 1996, 1116–1124.
• [18] Huang, Y., Benesty, J.: Audio Signal Processing for Next Generation Multimedia Communication Systems, Kluwer Academic Publishers, 2004.
• [19] Huang, Y., Benesty, J., Elko, G.: Passive acoustic source localization for video camera steering, Acoustics, Speech, and Signal Processing, 2000. ICASSP ’00. Proceedings. 2000 IEEE International Conference on, 2, 2000, ISSN 1520-6149.
• [20] Huang, Y., Benesty, J., Elko, G., Mersereati, R.: Real-time passive source localization: a practical linearcorrection least-squares approach, Speech and Audio Processing, IEEE Transactions on, 9(8), November 2001, 943 –956, ISSN 1063-6676.
• [21] Huang, Y., Benesty, J., G.Elko: Source Localization, chapter 9, Kluwer Academic Publishers, 2004, 229–253.
• [22] Kobayashi, K., Wynn, H.: Computational algebraic methods in efficient estimation, 2013.
• [23] Kraus, L.: A direct solution to GPS-type navigation equations, IEEE Transactions on Aerospace and Electronic Systems, AES-23(2), March 1987, 223–232.
• [24] Leva, J.: An alternative closed form solution to the GPS pseudorange equation, Proceedings of the Institute of Navigation National Technical Meeting, Anaheim, CA, January 1995.
• [25] Schau, H., Robinson, A.: Passive source localization employing intersecting spherical surfaces from time-of-arrival differences, Acoustics, Speech and Signal Processing, IEEE Transactions on, 35(8), August 1987, 1223 – 1225, ISSN 0096-3518.
• [26] Schmidt, R.: A New Approach to Geometry of Range Difference Location, Aerospace and Electronic Systems, IEEE Transactions on, AES-8(6), Nov. 1972, 821 –835, ISSN 0018-9251.
• [27] Siouris, G.: Aerospace Avionics Systems, Academic Press, San Diego, 1993.
• [28] Smith, J., Abel, J.: The spherical interpolation method of source localization, Oceanic Engineering, IEEE Journal of, 12(1), jan 1987, 246 – 252, ISSN 0364-9059.
• [29] Smith, J., Abel, J. S.: Closed-form least-squares source location estimation from range-difference measurements, IEEE Trans. Acoust., Speech, Signal Processing, ASSP-35, 1987, 1661–1669.
• [30] Spencer, S.: The two-dimensional source location problem for time differences of arrival at minimal element monitoring arrays, J. Acoust. Soc. Am., 121(6), June 2007, 3579–3594.
• [31] Spencer, S.: Closed–form analytical solutions of the time difference of arrival source location problem for minimal element monitoring arrays, J. Acoust. Soc. Am., 127(5), May 2010, 2943–2954.