Hyperbolic navigation refers to a class of
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 other form of navigation information can be used to eliminate this ambiguity and determine a fix.
The earliest known hyperbolic system was used during
World War I as an
acoustic location system for locating enemy
artillery. The sound of a shell being fired was received by several microphones, and the time of reception sent to a computing center to plot the location. These systems were used well into
World War II. By that time, however,
radio techniques were becoming much more capable, and most hyperbolic systems are based on radio means.
Basic conceptsTiming-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 3x10
8 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 had been carried out between 1935 and 1938 as part of the efforts to deploy
radar systems. The
UK, in particular, had invested considerable effort in the development of their
Chain Home system. The
radar display systems for Chain Home were based on
oscilloscopes (or oscillographs as they were known at time) triggered to start their sweep when the broadcast signal was sent. Return signals were amplified and sent into the 'scope display, producing a "blip". By measuring the distance along the face of the oscilloscope of any blips, the time between broadcast and reception could be measured, thus revealing the range to the target.
With very slight modification, the same display could be used to time the difference between two arbitrary signals. For navigational use, any number of identifying characteristics could be used to differentiate the master from the secondary signals. In this case, the portable receiver triggered its trace when it received the master signal. As the signals from secondary arrived they would cause a blip on the display in the same fashion as a target on the radar, and the exact delay between the master and secondary easily determined.
Hyperbolic navigation
Consider the same examples as our original absolute-timed cases. If the receiver is located on the midpoint of the baseline the two signals will be received at exactly the same time, so the delay between them will be zero. However, the delay will be zero not only if they are located 150 km from both stations and thus in the middle of the baseline, but also if they are located 200 km from both stations, and 300 km, and so forth. So in this case the receiver cannot determine their exact location, only that their location lies somewhere along a line perpendicular to the baseline.
In the second example the receivers determined the timing to be 0.25 and 0.75 ms, so this would produce a measured delay of 0.5 ms. There are many locations that can produce this difference - 0.25 and 0.75 ms, but also 0.3 and 0.8 ms, 0.5 and 1 ms, etc. If all of these possible locations are plotted, they form a hyperbolic curve centred on the baseline. Navigational charts can be drawn with the curves for selected delays, say every 0.1 ms. The operator can then determine which of these lines they lie on by measuring the delay and looking at the chart.
A single measurement reveals a range of possible locations, not a single fix. The solution to this problem is to simply add another secondary station at some other location. In this case two delays will be measured, one the difference between the master and secondary "A", and the other between the master and secondary "B". By looking up both delay curves on the chart, two intersections will be found, and one of these can be selected as the likely location of the receiver. This is a similar determination as in the case with direct timing/distance measurements, but the hyperbolic system consists of nothing more than a conventional radio receiver hooked to an oscilloscope.
Because a secondary could not instantaneously transmit its signal pulse on receipt of the master signal, a fixed delay was built into the signal. No matter what delay is selected, there will be some locations where the signal from two secondary would be received at the same time, and thus make them difficult to see on the display. Some method of identifying one secondary from another was needed. Common methods included transmitting from the secondary only at certain times, using different frequencies, adjusting the envelope of the burst of signal, or broadcasting several bursts in a particular pattern. A set of stations, master and secondaries, was known as a "chain". Similar methods are used to identify chains in the case where more than one chain may be received in a given location.
