Hyperbolic Navigation

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Hyperbolic navigation
Hyperbolic navigation refers to a class of radio navigation systems based on the difference in timing between the reception of two signals, without reference to a common clock. This timing reveals the difference in distance from the receiver to the two stations. Plotting all of the potential locations of the receiver for the measured delay produces a series of hyperbolic lines on a chart. Taking two such measurements and looking for the intersections of the hyperbolic lines reveals the receiver's location to be in one of two locations. Any form of other navigation information can be used to eliminate this ambiguity and determine a fix. The first such system to be used was the World War II-era Gee, introduced by the Royal Air Force for use by Bomber Command. This was followed by the Decca Navigator System in 1944 by the Royal Navy, along with LORAN by the US Navy for long-range navigation at sea. Post war examples including the well-known US Coast Guard LORAN-C, the international Omega system, and the Soviet Alpha and CHAYKA. All of these systems saw use until their wholesale replacement by satellite navigation systems like the Global Positioning System (GPS). Basic concepts

Timing-based navigation
Consider two ground-based radio stations located at a set distance from each other, say 300 km so that they are exactly 1 ms apart at light speed. Both stations are equipped with identical transmitters set to broadcast a short pulse at a specific frequency. One of these stations, called the "secondary" is also equipped with a radio receiver. When this receiver hears the signal from the other station, referred to as the "master", it triggers its own broadcast. The master station can then broadcast any series of pulses, with the secondary hearing these and generating the same series after a 1 ms delay. Consider a portable receiver located on the midpoint of the line drawn between the two stations, known as the baseline. In this case, the signals will, necessarily, take 0.5 ms to reach the receiver. By measuring this time, they could determine that they are precisely 150 km from both stations, and thereby exactly determine their location. If the receiver moves to another location along the line, the timing of the signals would change. For instance, if they time the signals at 0.25 and 0.75 ms, they are 75 km from the closer station and 225 from the further. If the receiver moves to the side of the baseline, the delay from both stations will grow. At some point, for instance, they will measure a delay of 1 and 1.5 ms, which implies the receiver is 300 km from one station and 450 from the other. If one draws circles of 300 and 450 km radius around the two stations on a chart, the circles will intersect at two points. With any additional source of navigation information, one of these two intersections can be eliminated as a possibility, and thus reveal their exact location, or "fix". Absolute vs. differential timing

There is a serious practical problem with this approach - in order to measure the time it took for the signals to reach the receiver, the receiver must know the precise time that the signal was originally sent. With modern electronics this is a trivial exercise, and forms the basis of all modern navigation systems, including GPS. In the 1930s, however, such precise time measurements simply weren't possible; a clock of the required accuracy was difficult enough to build in fixed form, let alone portable. A crystal oscillator, for instance, drifts about 1 to 2 seconds in a month, or 1.4x10-3 seconds an hour.[1] This may sound small, but as light travels 3x108 m/s, this represents a drift of 400 m per hour. Only a few hours of flight time would render such a system unusable, a situation that remained in force until the introduction of commercial atomic clocks in the 1960s. However, it was possible to accurately measure the difference between two signals. Much of the development of suitable equipment...
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