The number of global solutions for GPS source localization in two-dimension

TL;DR

It is proved that the number of possible two‐dimensional source locations with three measurements in L2 setting is at most 5, where the measurement triangle is isosceles and the measurement distance for the two isosceles triangles is the same.

Abstract

Source localization is widely used in many areas including GPS, but the influence of possible noises is not so negligible. Many optimization methods are attempted to alleviate different kinds of noises. Needless to say the stability of the solution, even the number of global solutions are not fully known. Only local convergence or stability for the optimization problem are known in simple $L^1$\cite{Kwon} or $L^2$\cite{Kwon3} settings. In this paper, we prove that the number of possible two dimensional source locations with three measurements in $L^2$ setting, is at most $5$, which is the complement and correction to the previous work \cite{Kwon3}. We also showed the sufficient and necessary condition for the number of the solutions being 1,2,3,4,and 5, where the measurement triangle is isosceles and the measurement distance for the two isosceles triangle bases are the same.

Figures (7)

• Figure 1: The figure for Theorem when $\left\lVert Z_1 Z_2\right\rVert=2$. The solutions are (a) $X=\{S_{12+}\}$ when $\left\lVert Z_1 Z_3\right\rVert<\left\lVert Z_1 Z_2\right\rVert$, (b) $X=\{S_{12+}\}$ when $\left\lVert Z_1 Z_3\right\rVert<\left\lVert Z_1 Z_2\right\rVert$ and $d_1<P$, (c) $X=\{ S_{23\pm}, S_{31\pm}\}$ when $\left\lVert Z_1 Z_3\right\rVert<\left\lVert Z_1 Z_2\right\rVert$ and $d_1>P$, and (d) $X=\{ S_{12+}, S_{23\pm}, S_{31\pm}\}$ when $\left\lVert Z_1 Z_3\right\rVert<\left\lVert Z_1 Z_2\right\rVert$ and $d_1=P$ .
• Figure 2: The figure for (a) Lemma and (b) Lemma .
• Figure 3: The figure for Lemma .
• Figure 4: Source locations when the measurement triangle is equilateral with $r=2$ (Theorem ) and when $(d_1, d_3) =$ (a) $(1.3333,1.9737)$ (b) $(2.6000,1.3000)$ (c) $(2.6000,2.6000)$ (d) $(4.0000,4.4495)$ (e)$(4.000,4.8990)$ (f)$(4.000,5.2520)$.
• Figure 5: Solution positions for flat isoscleles $(s<\frac{\sqrt 3 r}{2})$ with $s=1$ and $r=2$ in Theorem , when $(d_1, d_3)=$ (a) $(1.8251,1.7725)$ (b) $(1.8251,1.9204)$ (c) $(2.2361, 1.2477)$ (d) $(2.2361,2.2361)$ (e) $(4.4721,3.9155)$ (f) $(4.4721,4.4721)$.

Theorems & Definitions (27)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof