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I am trying to develop an idea to determine the location of a cell phone using OTDOA (obseved time differences of arrival). The base stations emit electromagnetic wave to the cell phone, the cell phone records the time differences(OTDOAs) of different base stations, for example, $T_1$ is the time difference of station $1$ to station $0$, $T_2$ is the time difference of station $2$ to station $0$.

Here's what we know: $X_i,Y_i$ of $3$ to many base stations, where $X_0,Y_0$ is the reference station. $T_i$ of OTDOA for station $i$ to reference station $0$. $C$ is the speed of light or electromagnetic wave. We only do the computation on a plane, no $Z$ needed.

Here's what we want: $x,y$ of the cell phone.

This eventually comes to two (or more) branches of two (or more) hyperbolas, because for every $T_i$ we can draw a branch of a hyperbola.

How can I compute the $x,y$ for the cell phone with known information? Thank you so much!

May be useful: Intersection of two hyperbolas

Equation of one branch of a hyperbola in general position

tarit goswami
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baisong
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  • Welcome to math.SE. You should use MathJax/LaTeX to format your post, it makes it much more readable, the help center has some links to get you started. You should include more of your own thoughts/work, for instance: why do we get a hyperbola for each $T_i$? – Henrik supports the community Aug 13 '18 at 06:10
  • Why do you want to use herperbola's ? Otherwise, the problem is "quite" simple usinng cartesian coordinates.. – Claude Leibovici Aug 13 '18 at 06:18
  • Hi Henrik, will try to use MathJax/LaTaX next time. – baisong Aug 13 '18 at 06:44
  • Hi Claude, Could you provide more details, it's a big headache for me. The reason I am using hyperbola is every Ti leads to a branch of a hyperbola, because the distance difference is constant = Ti * C, this leads to a branch of a hyperbola. – baisong Aug 13 '18 at 06:47
  • @baisong I have edited the post and Latexed it,please accept the edit. Can you explain it more? Means, are they special type of hyperbola? Can you explain the mathematics behind it in more details? – tarit goswami Aug 13 '18 at 07:14
  • Are you looking for an algorithm that doesn’t require solving any equations? There’s a general one for computing the intersection of two conics that could certainly applied to your problem. Unfortunately, you’re working with real-world data, so it’s highly unlikely that the collection of hyperbolas you derive will have a neat common intersection. You’re almost certain to end up having to estimate the intersection point somehow. – amd Aug 13 '18 at 18:51
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    @taritgoswami Thanks for the editing. According to the definition of hyperbola, we can see that it's a collection of points that have a constant distance difference between focus A and B (in my example they are reference station 0 and anther station i), i.e., distance(cell phone to station 0) - distance(cell phone to station i) = constant = 2a, where a is a parameter of the hyperbola equation. – baisong Aug 14 '18 at 05:18
  • @amd I am looking for a way to compute the intersection(s), no limited in solving its equations however I do think this is the best way to find the intersections (I think one branch of a hyperbola intersects another branch of another hyperbola can have possible 0, 1, 2, 4 intersections). – baisong Aug 14 '18 at 05:21

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