# What the Heck is a Pulsed Hyperbolic System Anyway?

Finding our way to any destination is something we take for granted today with global positioning system (GPS) apps. But we didn’t always have GPS. What did we do before?

In 1920, the first aircraft experimented with radio navigation. A US naval seaplane took off from Norfolk VA to see if it could find a radio transmitter onboard the U.S.S. Ohio. The ship was at sea nearly 100 miles away, but it only took the plane five minutes to establish a heading towards the ship. 90 minutes later, it circled the ship, and started to head home.

And how do you suppose the plane found its way home? It followed a different radio signal, a local AM radio station in Norfolk. You can read more about this story at https://www.wired.com/2010/07/0706aircraft-radio-compass/

This idea of radio direction finding was formalized by the military during World War II. It was initially called LORAN, which stood for Long Range Navigation. It was a land-based system which used radio signals to help US military planes and ships within 600 miles of the coast navigate. This eventually became known as Loran-A as new generations of loran came along.

A more accurate loran system, Loran-C, was developed in the 1950s for civilian use. Loran-C used the 90-110 kHz frequencies, and had a range of over 2000 miles. It was eventually extended to cover most of the continental United States, and Loran-A was phased out. Loran-C became very popular worldwide, but was supplanted by GPS.

How does loran work? The idea is pretty simple. Suppose you are a ship offshore trying to navigate using loran. You receive radio signals from three loran stations – a primary station A and two secondary stations B and C. This collection of stations works together. The primary station broadcasts regular pulses at a fixed rate and duration. The secondary stations also transmit pulses, each at its own fixed offsets to station A’s pulses. All the stations are 200-300 miles away from one another, and their locations are known.

Now remember this is a system designed before GPS could provide an accurate time signal to equipment. The clock on the ship is not synchronized with stations A or B’s clocks. But the ship can still navigate by using the elapsed time between when signal A and signal B were received. This is known as the time difference of arrival (TDOA) between signals A and B.

Suppose that station A sends a signal at time t_{A}, and it takes delta(t_{A}) to get to the ship, where we don’t know what t_{A} or delta(t_{A}) are. Station B will send its signal at a known time interval k_{1 }after station A sends its signal. Station B will transmit its signal at time t_{B }and the signal takes delta(t_{B}) to get there. Again, on the ship we don’t know what t_{B} or delta(t_{B}) are.

But we can measure the TDOA between when signals A and B were received on the ship. Noting that t_{B} = t_{A} + k_{1,} we can write the TDOA as:

(t_{B} + delta(t_{B})) – (t_{A} + delta(t_{A})) = (t_{A} + k_{1} + delta(t_{B})) – (t_{A} + delta(t_{A})) = k_{1} + delta(t_{B}) – delta(t_{A})

I already know k_{1}, so now I have an accurate TDOA estimate delta(t_{B})_{ }– delta(t_{A}). Since radio waves travel at the speed of light L, I can write the difference in the distance of the ship from stations A and B as

L* (delta(t_{B})_{ }– delta(t_{A})) = L*delta(t_{B})_{ } – L *delta(t_{A}) = d_{B} – d_{A}

where d_{A} is the distance between station A and the ship, and d_{B} is the distance between station B and the ship.

Now look at the attached diagram. A hyperbola is defined as the set of all points where each point P is a constant difference D in length from its two focal points F_{A} and F_{B }. This difference D, our d_{A} – d_{B}, is actually equal to +/- 2a, where a is the length of the hyperbola’s semi-major axis. You can prove this by looking at the diagram and using the Pythagorean theorem:

d_{A} – d_{B} = sqrt [(x_{1} +c)^{2} + y^{2}] – sqrt [(x_{1}– c)^{2} + y^{2}] = D (1)

Multiply numerator and denominator by sqrt [(x_{1} +c)^{2} + y^{2}] + sqrt [(x_{1}– c)^{2} + y^{2}] and rearrange:

4xc/D = sqrt [(x_{1} +c)^{2} + y^{2}] + sqrt [(x_{1}– c)^{2} + y^{2}] (2)

If we add equations (1) and (2), and substitute in the value for c = sqrt (a^{2} + b^{2}), we can show that D = +/- 2a.

So the TDOA associated with d_{B} – d_{A} can tells us which hyperbolic curve we are on. There are actually two pieces to the curve, shown as the green curve and the purple curve in the diagram. These correspond to the +2a and -2a differences. Luckily, by adding one more station C and calculating its TDOA curves with Station A, we’ll find a place where these hyperbolic curves intersect the original curves on one place. This common point of intersection gives us a fix on our location. This is why Loran-C is called a pulsed, hyperbolic system.

Today, GPS has largely displaced other methods of navigation, but radio direction finding still can provide a valuable backup system should GPS become unavailable or unusable. And that is why new life has been breathed into loran technology with Loran-E, Enhanced Loran.

I’ve repeatedly referred to GPS in today’s blog, without talking about how it really works. The time has finally come to talk about this technology, which we will do next week.