TDOA Geometry

Situation

An emitter is transmitting radio frequency (RF) energy. We have no way of decoding its signal but we want to find out where it is coming from. Time of arrival (TOA) systems can be very precise, but they can only be used when the system has knowledge of when the transmission was sent. This is the case with GPS, where each signal contains information that says "I sent this signal at xx:xx:xx time". With TOA off the table, we can use either multiple distributed angle of arrival (AoA) systems, a distributed time difference of arrival (TDOA) system, or a combination of both. AoA needs directional antennas or an interferometric array and has the drawback of getting worse at long ranges. TDOA needs synchronized clocks and is highly dependent on geometry.

Because implementing TDOA has a simpler antenna baseline, this article will go over the basics of 2-dimensional, 3 receiver TDOA system geometry.

The TDOA Isochrone

To locate an emitter with TDOA, we first measure the time difference between the received signals, $\tau$. Then, because RF propagates at a constant speed in a given medium, we can calculate the range difference, $d$, of the emitter by multiplying $\tau$ by the speed of light, $c$.

$$d = c\tau$$

The set of points whose range difference has magnitude $|d|$, is a hyperbola with the pair of receivers as the foci. For one pair of receivers, we get one hyperbola on the map. when you take into account the sign of $\tau$ (which receiver saw the signal first), we can reduce that to just one branch of the hyperbola. The emitter can be anywhere along that branch, which we call an isochrone.

Three Receivers Produce a Fix

With the third receiver, we produce another isochrone. The intersection of the two isochrones is the candidate position of our emitter! In general, $N$ receivers give $N-1$ independent measurements. To get a 2D fix, we need 3 receivers. For a 3D fix, we would need 4 receivers. But beware, with only 3 receivers, it is possible that the isochrones cross at two separate points, which is an ambiguity we need to solve. We can shake off the false fix with another receiver or some transmitter or receiver motion over time.

Click anywhere in the box to add an emitter. The two blue curves are its isochrones and the green circles are the TDOA receivers. You can drag the receivers around to change the geometry of the TDOA system too.

1. Start inside the receiver triangle. You'll see the isochrones intersect perfectly over the emitter.
2. Drag a receiver around. See the isochrones change with geometry.
3. Move the emitter far outside the triangle. Now the curves cross at a shallow angle. Imagine what might happen to the intersection if one of the curves were moved a little bit. It would result in a big geolocation error.
4. Try to make the curves intersect twice. If you can do this, you can see where ambiguities might appear in a 3 receiver system.

What's Left

The geometry of TDOA is the most visible part to a typical operator, but it is not the whole story. We still need to learn a few things like how to measure the time delay, how to visualize error in our time delay measurement, how often to update the isochrones, how to resolve ambiguities, and what about using a heat map instead of isochrones? There is a lot more to be said about TDOA, which we will save for another write up!

Thanks for reading!