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On the Uniqueness of Solutions in GPS Source Localization: Distance and Squared-Distance Minimization under Limited Measurements in Two and Three Dimensions

Kiwoon Kwon

Abstract

The source localization problem, fundamental to applications like GPS, is typically approached as a minimization problem in the presence of various types of noise. Ensuring the uniqueness of solutions in GPS technology is vital for the reliability and accuracy of applications, from everyday navigation to critical military operations. In this paper, we examine two key minimization problems: one focused on distance error and the other on squared distance error. We explore these problems in both three-dimensional space, the standard scenario, and in two-dimensional space as a simplified case. Furthermore, we discuss the number of possible source solutions when the number of measurements is fewer than three.

On the Uniqueness of Solutions in GPS Source Localization: Distance and Squared-Distance Minimization under Limited Measurements in Two and Three Dimensions

Abstract

The source localization problem, fundamental to applications like GPS, is typically approached as a minimization problem in the presence of various types of noise. Ensuring the uniqueness of solutions in GPS technology is vital for the reliability and accuracy of applications, from everyday navigation to critical military operations. In this paper, we examine two key minimization problems: one focused on distance error and the other on squared distance error. We explore these problems in both three-dimensional space, the standard scenario, and in two-dimensional space as a simplified case. Furthermore, we discuss the number of possible source solutions when the number of measurements is fewer than three.
Paper Structure (12 sections, 12 theorems, 33 equations, 7 figures, 4 tables)

This paper contains 12 sections, 12 theorems, 33 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. One measuremt (I=1)
  3. Two measuremts (I=2)
  4. The squared distance error case ($\psi(y)=y^2$)
  5. The distance error case ($\psi(y)=y$)
  6. The comparison
  7. Three measuremts (I=3)
  8. The squared distance error case ($\psi(y)=y^2$)
  9. The distance error case ($\psi(y)=y$)
  10. The comparison
  11. Conclusion
  12. Acknowledgements

Key Result

Lemma 1

Under the condition that the following sets are nonempty in $R^n, n=2,3$, we have (a) The minimum in $(D_1 \cup D_2)^c$ is the point nearest to $Y_0$. (b) The minimum in $D_1\cap D_2$ is the farest point to $Y_0$. (c) The minimum in $D_1\setminus D_2$ is $N_1$ if $N_1\notin D_2$ and $S_{12}$ if $N_1

Figures (7)

  • Figure 1: (a) The relation between the distance to $Z_1$ and the objective function is shown for both the distance error (blue line) and the squared distance error(red line). (b,c) The level sets of the objective function are depicted for $Z_1= O$ and $d_1=1$ in the $xy$-plane.The red line is the level set $0$ and the blue lines are level sets for $0.5, 1, 1.5,$ and so on. A three dimensional sphere is plotted with blue dots. In (d) and (e), the values of level sets are displayed in $xy-, yz-, zx-$ planses. In (f) and (g), the streamlines and the gradient vectors are shown. Specifically, (b),(d), and (e) corresponds to the distance error case, while (c),(d), and (g) represent the squared distance error case.
  • Figure 2: The location of the two-diemsional sources for (a,c,e,g) $\psi(y)=y$ and (b,d,f,h) $\psi(y)=y^2$ when (a,b) $d_1\le \frac{\left\lVert Z_1 Z_2\right\rVert}{2}$ (c,d) $\frac{\left\lVert Z_1 Z_2\right\rVert}{2}\le d_1 <\left\lVert Z_1 Z_2\right\rVert$ and $d_1+d_2\le\left\lVert Z_1 Z_2\right\rVert$ (e,f) $d_1 - d_2 < \left\lVert Z_1 Z_2\right\rVert < d_1 + d_2$ (g,h) $d_1 - d_2 \ge \left\lVert Z_1 Z_2\right\rVert$
  • Figure 3: The location of the three-dimensional sources for (a,c,e,g) $\psi(y)=y$ and (b,d,f,h) $\psi(y)=y^2$ when (a,b) $d_1\le \frac{\left\lVert Z_1 Z_2\right\rVert}{2}$ (c,d) $\frac{\left\lVert Z_1 Z_2\right\rVert}{2}\le d_1 <\left\lVert Z_1 Z_2\right\rVert$ and $d_1+d_2\le\left\lVert Z_1 Z_2\right\rVert$ (e,f) $d_1 - d_2 < \left\lVert Z_1 Z_2\right\rVert < d_1 + d_2$ (g,h) $d_1 - d_2 \ge \left\lVert Z_1 Z_2\right\rVert$.
  • Figure 4: The source locations for cases (a,c,e,g) are based on $\psi(y)=y$, while those for cases (b,d,f,h) are based on $\psi(y)=y^2$, The conditions for each case are (a,b) $d_1\le \frac{\left\lVert Z_1 Z_2\right\rVert}{2}$ (c,d) $\frac{\left\lVert Z_1 Z_2\right\rVert}{2}\le d_1 <\left\lVert Z_1 Z_2\right\rVert,d_1+d_2\le\left\lVert Z_1 Z_2\right\rVert$ (e,f) $d_1 - d_2 < \left\lVert Z_1 Z_2\right\rVert < d_1 + d_2$ (g,h) $d_1 - d_2 \ge \left\lVert Z_1 Z_2\right\rVert$. For detailed description of these cases see Tab. .
  • Figure 5: The source locations for cases (a,c,e,g) are based on $\psi(y)=y$, while those for cases (b,d,f,h) are based on $\psi(y)=y^2$, The conditions for each case are (a,b) $|T|=2$ and (c,d) $|T|=1, N_1\in D_2$. For detailed description of these cases see Tab. .
  • ...and 2 more figures

Theorems & Definitions (15)

  • Lemma 1
  • proof
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Lemma 5
  • proof
  • Theorem 6
  • Corollary 7
  • Corollary 8
  • ...and 5 more