Math

[mojoe]

I have in mind a project that involves Time Difference of Arrival (TDOA). I have looked at several papers that discuss this and understand that it involves hyperbolic curves created from pairs of fixed points to find the location of the unknown source (whether it be sound or radio waves).

My problem is that my math in this area is weak. Many years ago, I did very well with Algebra, Geometry and Trigonometry. I still retain and occasionally use a bit of Algebra and Trig.

I'd like recommendations on study material to bring me up to speed on how to do the required math for TDOA calculations. Ultimately, I will probably code things in Python, which has quite a selection of math libraries. However, I want to know how to do the math myself, first.

Study material that I can download and use offline is preferred. A mixture of ebooks and videos would be great (I can download Youtube videos and some others). In the coming months, I will be without internet for long stretches.

When I finish my project, I will have the GPS coordinates of four or more fixed ground stations, each one reporting a GPS-derived timestamp of when it received a signal. I would like to plot the hyperbolic curves on a map (prehaps Google Maps) and show the center point of the overlapping curves, to indicate the source location.

Thanks.

[Benta]

Look at LORAN, the system used everywhere in shipping before GPS. You'll find lots of material.

[JimRemington]

Look up trilateration and multilateration. Here are a couple of places to start:
https://en.wikipedia.org/wiki/Trilateration
https://en.wikipedia.org/wiki/Multilateration

[mojoe]

Yes, thanks for the links (and the mention of LORAN), but I have read about those. That still doesn't help me with doing the math.

[gnif]

This might help

[cdev]

Check out http://en.blitzortung.org . Its a project which uses precise GPS timing and TDOA to map lightning strikes in real time all around the world.


For example, here is a realtime lightning map of the US.

http://en.blitzortung.org/live_lightning_maps.php?map=30

You could probably learn a lot about how it works from examining the code.

Also, there are lots of math classes online at sites like Archive.org, Khan Academy,  and even youtube.


Note that the frequency of the signals is going to make a very great deal of difference in how (and if) it propagates beyond the horizon, especially how long it takes.

For example, some signals (HF) literally go thousands of miles into space before they bounce back to earth.  Other signals (VHF+UHF) might propagate locally by line of sight but rarely encounter conditions which allowed them to go farther than the horizon. Also, signal propagation conditions change through the day and night and seasonally, also there is an eleven year sunspot cycle.

[Mattjd]

Going out on a limb here, because I haven't study signals yet, but to me it sounds like what you want is hyperbolic geometry. If you did well in geometry, assuming it was taught correctly i.e. axiomatically, then you should be able to learn it. The difference between euclidean geometry and hyperbolic is the changing of Euclid's 5th postulate. The parallel postulate. You can change it two ways, one way gives rise the hyperbolic geometry, the other is elliptic geometry. 

See parallel postulate section in the link below.

https://en.wikibooks.org/wiki/Geometry/Hyperbolic_and_Elliptic_Geometry

Hyperbolic geometry is the math used in GPS systems. You know how in movies when they're tracking crap via GPS and they're all like "wait while I triangulate the coordinates." Well that is exactly what happens. Imagine a globe of Earth, a triangle is drawn on it and that is the triangulation being spoken of. However, it is not a planar triangle like one would study in Euclidean geometry and used in Trigonometric. I.e. the sum of the internal angles do not equal 180 degrees. What is really placed on the globe of the Earth is a hyperbolic triangle. Totally different kind of math.

[brainwash]

Cramer's rule will help to approximate the polynomial equation from a given set of points. That's how I can read your initial request - I think you are trying to fit a curve onto some points. If it's not polynomial, you might want to read on curve fitting.
Signal theory only comes up to a point, afterwards it's all geometry, no need for fancy math.
If you are using phased arrays elements it would be a bit more complicated, but still resolves to pretty similar approximations.

[mojoe]

Thanks for the helpful replies. I'll take a look at them.

Brainwash - I will need to calculate a set of hyperbolic curves, based on each pair of receiving stations, which will have known, fixed coordinates (lat/long). With four receiving stations, that will be six unique pairs of coordinates. Each receiving station will also report a time stamp of when it received the signal in question. Among other uses, I understand that this is used to locate gun shots, and it was how your cell phone was located, prior to every phone having GPS.

[brainwash]

Excuse my ignorance as I haven't tackled this exact problem before, but why hyperbolic? I assume the stations sending the time are within LOS (<30km) so the distance can be approximated as a linear vector. Prior to GPS a lot of [civil] positioning was done in 2d, as far as I recall. If 3D positioning is required then I assume the timing constraints go up exponentially.

[mojoe]

Don't ask me. I am just going by what information and academic papers I have read. Perhaps someone who understands this better will explain.

Edit: Section 3.3 of this paper explains it the best I've seen so far (thanks cdev).
http://en.blitzortung.org/Compendium/Documentations/Documentation_2014-05-11_Red_PCB_10.4_PCB_12.3_PCB_13.1_PCB_14.1.pdf?1501388473

[nfmax]

I have in mind a project that involves Time Difference of Arrival (TDOA).
...
When I finish my project, I will have the GPS coordinates of four or more fixed ground stations, each one reporting a GPS-derived timestamp of when it received a signal. I would like to plot the hyperbolic curves on a map (prehaps Google Maps) and show the center point of the overlapping curves, to indicate the source location.

Let me guess - passively locating nearby aircraft by recording the times of arrival of their transponder transmissions? 

[mojoe]

No, not aircraft. What I have in mind is jammers on Amateur VHF/UHF.