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Two-Dimensional Frequency-Difference-of-Arrival Varieties

Jeanne Duflot, Margaret Cheney, James A. Given

Abstract

This paper studies Frequency-Difference-of-Arrival (FDOA) curves for the 2-dimensional, 2-sensor case. The primary focus of this paper is to give a description of curves associated to the FDOA problem from the algebro-geometric point of view. To be more precise, the complex projective picture of the family of FDOA curves for all possible relative velocities is described.

Two-Dimensional Frequency-Difference-of-Arrival Varieties

Abstract

This paper studies Frequency-Difference-of-Arrival (FDOA) curves for the 2-dimensional, 2-sensor case. The primary focus of this paper is to give a description of curves associated to the FDOA problem from the algebro-geometric point of view. To be more precise, the complex projective picture of the family of FDOA curves for all possible relative velocities is described.
Paper Structure (63 sections, 28 theorems, 320 equations, 5 figures)

This paper contains 63 sections, 28 theorems, 320 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem formulation and strategy
  3. Notation and geometry
  4. TDOAs
  5. Doppler shifts and FDOA curves
  6. Strategy and formulation of the modified FDOA problem
  7. Use of projective space
  8. Projective varieties
  9. Using algebraic geometry to study equations involving Euclidean norms
  10. FDOA conditions in complex projective space
  11. The FDOA problem: Summary
  12. Different coordinate systems
  13. The "original" coordinates
  14. The $[\bm w,x_1]$-coordinates
  15. The $[\bm z,x]$-coordinates
  16. ...and 48 more sections

Key Result

Lemma 20

The projection of real varieties $\Pi:Y(Q,Q_1) \rightarrow Y(Q)$ defined (everywhere) by gives $Y(Q,Q_1)$ as a double cover of the smooth variety $Y(Q)$ branched over the reduced and reducible curve $L$ defined by $w_0w_3=0$.

Figures (5)

  • Figure :
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  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (64)

  • Remark 8
  • Remark 17
  • Lemma 20
  • proof
  • Theorem 23
  • proof
  • Remark 30
  • Lemma 32
  • proof
  • Corollary 33
  • ...and 54 more