Communications in Information and Systems

Volume 12 (2012)

Number 2

Polynomial calculations in Doppler tracking

Pages: 157 – 184

DOI: http://dx.doi.org/10.4310/CIS.2012.v12.n2.a2

Authors

Tsung-Lin Lee (Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan)

Song-Sun Lin (Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan)

Wen-Wei Lin (Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Ubo Zhu (College of Science, National University of Defence Technology, Changsha, China)

Abstract

Tracking a moving object by the Doppler effect is an important tool to locate the position and to measure the velocity of a moving object. In theory, the corresponding movement of the object can be formulated by a system of 12 quadratic polynomials in 12 unknowns. In this paper, we mainly propose a novel simplification to reduce the original system to a system of 4 polynomials of degrees 4, 3, 2 and 2 in 4 unknowns. Furthermore, we can also reduce the original system to a new system of only three quadratic polynomials in 3 unknowns when the 6 observation stations are located at the vertices and centre of a regular pentagon, numerical experiments show that the simplified polynomial system can be solved by the homotopy method efficiently and reliably. The method is much more robust than Newton’s method when the initial vector is far from the solution. Also, the regular pentagon case outperforms the other configurations in terms of numerical accuracy.

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