Hyperbolic navigation is a means of navigation which works by measuring the difference in arrival time between two signals. This may be done by measuring either time-of-arrival or phase-difference. Phase difference is usually preferred as it removes the need to precisely synchronise many transmitting stations.
A chain of transmitters consists of one or more transmitters. Generally there will be one master station which sets the timing for an entire chain, and a constellation of slave stations which repeat the master (or another station's) signal.
A receiver needs to receive signals from several transmitter pairs in order to obtain a location fix.
Phase-difference systems (PDOA) operate by measuring the phase of a slave signal against its designated master. The master-slave relationships form a series of baselines, which are drawn directly from the master to the slave. At the centre of the baseline (equidistant between the two transmitters) and several points along the baseline, the phase difference between master and slave is zero, i.e. the signals are in phase. These points, when drawn out to the radius the transmitters cover, form hyperbolic curves on the map. These are called Lines of Position (LOPs).
A lane is the distance between two LOPs on the baseline. This is equal to the wavelength of the transmitted signal.
Several phase measurements are taken and the points of intersection of the respective LOPs is found. For a single-frequency system, these will generally not result in an unambiguous fix. That is to say, there will be multiple points where the same phase difference could have been observed. This leaves one remaining problem: how to identify which of the position fixes is correct.
Generally this is done by using a lower-frequency “coarse” position fix signal. The same calculations are done to obtain a position fix, but the longer wavelength increases the distance between lanes. At the coarse-acquisition frequency used in Datatrak (80Hz), the lane distance is 3747km, or 2329 miles. This is compared to the 23km (14.3 miles) lane width of the prime difference signal (13kHz difference frequency), or the 2.3km (1.4 miles) lane width of the 130kHz signal.
Datatrak follows the master/slave model, but uses two fixed-frequency chains split into slots in the time domain.
A master station generates the synchronisation data and one timing slot for the chain it operates on. Each slave station receives the phase signal from the master, and repeats it (i.e. operates as a phase mirror). This is very similar to how Decca functions – except that the receiver only needs one LF receiver (Decca requires three receiver-multiplier and phase-comparator blocks).
There are two chains: “$F_1$” (first frequency) which covers areas south of the Midlands; and “$F_2$” which covers Northern England and Scotland.
A “switching line” determines which of these is used: North of this line, the $F_2$ chain is used. South of the line, the $F_1$ chain is used.
The notation indicates where the master timing data (which the receiver synchronises to) is received from – both chains are in synchronisation and transmit the same data.
This is the simplest possible Datatrak signal.
The signal for the F1 chain (for basic Datatrak) looks like this:
| $F_1$ | Sync and timing | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $F_2$ | … | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
This only allows for a single chain, and is only used for illustration.
The receiver (also known as a Locator) tunes to $F_1$ to receive the sync and timing data and make phase measurements of the eight navigation slots against the receiver's internal temperature-compensated oscillator. The receiver then switches to $F_2$ and makes the same measurements on the F2 frequency.
Each navigation slot is 80ms long – with 40ms transmitted at a higher frequency ($F_1+$ and $F_2+$) and 40ms at a lower frequency ($F_1-$ and $F_2-$).
The basic signal encoding is fine, but the limit of eight slots restricts the number of transmitters which can exist in the network. It's obvious from the timing diagram above that only half of the transmitter capacity is being used. This is easily fixed:
| $F_1$ | Sync and timing | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $F_2$ | Sync and timing | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
This allows us to have 16 transmitters, with one caveat. As the Locator only has one receiver, it can only tune to one frequency at a time. This prevents the receiver from receiving slots 1 and 9 at the same time. This is called a slot collision.
When planning the network, care must be taken to keep slot collisions to a minimum as Locators move around the signal area.
Interlacing can be extended further by linking two 1.68-second cycles into a larger cycle pair:
| $F_1$ | Sync and timing | 1..8 | 9..16 | Sync and timing | 1..8 | 17..24 |
|---|---|---|---|---|---|---|
| $F_2$ | Sync and timing | 9..16 | 1..8 | Sync and timing | 17..24 | 1..8 |
This expands the system to a maximum of 24 navigation slots, with a further caveat: the measurements for slots 1 to 8 are updated every 1.68-second cycle, but the measurements for slots 9 to 16 and 17 to 24 are updated every two cycles (3.36 seconds).
Slot collisions are still an issue, but the rules change slightly:
(See also Hoffman-Wellenhoff et al, 2003)
Datatrak uses a three-step process to obtain a position fix:
This system has one massive advantage: the entire navigation solution can be obtained in three stages from the same input data.
Once the position of the receiver is known, successive positioning calculations may only need the “fine” adjustment (provided it can be assumed that the vehicle has not crossed a lane during successive phase measurements).
This works because a phase measurement at one frequency subtracted from a phase measurement at another frequency will result in a phase measurement taken at the difference in frequency between the two signals – 80Hz in the case of the super-coarse fix.
To put this in perspective, the relative accuracies of the different phases are:
| Phase | Frequency | Wavelength (lane width) | Resolution |
|---|---|---|---|
| “super-coarse” | $\Delta F = 80 \mathrm{Hz}$ | $\lambda \approx 3747 \mathrm{km} (2329 \mathrm{miles})$ | $\frac{3747 \mathrm{km}}{1000} \approx 3.7 \mathrm{km}$ |
| “coarse” | $\Delta F \approx 13 \mathrm{kHz}$ | $\lambda \approx 23 \mathrm{km} (14.3 miles)$ | $\frac{23 \mathrm{km}}{1000} \approx 23 \mathrm{metres}$ |
| “fine” | $F \approx 130 \mathrm{kHz}$ | $\lambda \approx 2.3 \mathrm{km} (1.43 miles)$ | $\frac{2.3 \mathrm{km}}{1000} \approx 2.3 \mathrm{metres}$ |
In practice, the total accuracy is limited by variations in propagation conditions and measurement error.