Fundamental Methodology of Signal time of Flight for GPS


Importance of time intervals

This Post was borrowed from LAND SURVEYORS UNITED, submitted by William H. Burling. This explanation works if the effects of a moving satellite are neglected.  That factor will be addressed at the end.

Background:   “time intervals” and “time”

When talking about time in GPS, one must differentiate between “time intervals” and “time”.

Time intervals (such as seconds in a clock) are not time.  An interval of time can be described by an event and wait period until the next event appears.   There is no summing of intervals.

Time is the summation of pulses (or intervals) from some agreed upon reference. 8 pm is 60 seconds after 7:59 pm. Or 7.2E4 seconds from midnight.  Or Y seconds from some reference years ago. The reference can be anything and is only important for the situation of interest.  The most important point to understand with time is that it is always focused on the summation of pulses.

GPS and time intervals

GPS is fundamentally based on a signal consisting of a sequence of pulses that are identical in pulse width and pulse period (time between pulses) both at the satellite and at the receiver.

Initially, when a receiver is turned on, the pulse train coming from the satellite and the pulse train from the receiver  are independent. They approximately have the same pulse widths and frequency but are not in phase.

The pulse trains at the receiver are guaranteed to replicate the pulse trains from the satellite by using a phase lock loop (PLL) syncing methodology. Thus the two pulse sequences (one from the satellite and one at the ground station) appear to have the exact same frequency and the pulse trains are in phase.

If presented visually, one would see a satellite created pulse directly above and aligned (PLL) with a pulse created in the receiver.

But the satellite pulse directly above and aligned with the receiver pulse was made much earlier.

Satellite signals do not travel instantly. Signals are delayed x seconds for every y miles.

Keeping in mind that for the moment we are talking about a non-moving satellite for this explanation, it does not matter that the satellite’s pulses may have been made earlier than the pulses created in the receiver. The pulse train proceeds without any impact on the signal pulse or its frequency (such as Doppler) until it finally arrives intact wherein the receiver pulse train is synchronized.

Looking for a time reference

If one could magically, in a single reference time, identify one of the pulses at the satellite and one of the pulses at the receiver, then when the pulse from the satellite arrived at the receiver, the two marked pulses would be displaced from one another in integral pulse units which can be used to determine the flight time of the pulse from satellite to the receiver.  Knowing flight time AND the time the flight began (this number is only  involved in computing satellite position) enables distance to the known position of the satellite to be calculated. Note that the resolution of known distance is limited to the time interval between two adjacent pulses.

So the question is how does the magic occur in emulating a single reference time?

For the satellite to know reference time (defined on earth), it has to be accomplished by knowing exactly how far the satellite is from some known time reference on earth so that when it receives a “time stamp”  the distance is taken into consideration along with all kinds of second and third order effects. This is currently achieved by using a ground station that takes into consideration the delays in getting the reference time from a reference located on the globe to the ground station and then knows the distance from ground station to the satellite. It is a two-step process but the two step process is able to better accommodate for all the factors that tend to degrade the transport of the reference time.

For the receiver it is not so easy to know reference time.  It can’t get it from the satellite as it does not know how far away the satellite is and the receiver does not know how far away it is from the earth reference time station.

Reference time for the receiver is derived, not given

It turns out that the receiver does not need to know a single reference time. The explanation based on a single reference time only enables people to appreciate that time of flight can be the basis for GPS.

Instead, the receiver only needs to know that its location is at one and only one place. Sounds ridiculous, but this is how I believe it works:

The receiver guesses which of its pulses is the one that reference time would have selected. With that single guessed pulse, it concurrently guesses the receiver’s distances from at least four satellites and their known locations.

Any three distances define a location through trilateration. Thus having access to X satellites (equal to or greater than four) the receiver can calculate Y distances and as many locations, where Y is the number of combinations of X satellites taken 3 at a time (xC3)

The initial guess as to which pulse is the reference is likely to result in xC3  locations being predicted.

So the single guessed pulse is shifted to a new guess pulse resulting in a new calculated time of flight for each of the pulse trains from each of the satellites.  The xC3 time of flight guesses produces xC3 guessed distances and xC3 new guessed location.

Guesses continue to change until all Y distances result in a single location being predicted. Of course guessing is enhanced in several ways not of importance here.

Notice that the final guess of the reference pulse that results in a single location IS the delay that occurred when the reference pulse was issued from the satellite but the receiver did not need to know what that value was before it calculated the location of the receiver.

The reference time that is part of a message sent in the pulse train from the satellite is important but is not used in the time of flight calculation. Its importance is in allowing the receiver to look up a satellite location based on its unique id and the reference time sent with the pulse train. That satellite location is critical to the calculation of the distance between satellite and receiver.

Syncronizing of all satellites and ground stations

All of the Satellites can be synced to one Reference time as a fixed ground station knows about its delays getting the reference time from one part of the earth to its location and the ground station knows the exact distance from the ground station to each satellite when it declares a Reference time. Syncing means the” marked” pulses I mentioned above are associated with the Reference time. Marked pulses in GPS terminology is actually a sequence of pulses called a Pseudorandom Noise Number (PRN), but knowing about PRN is not important here. Its primary function is to be a time marker and for an explanation, the time marker can be a single pulse.

Syncing also involves another process not relevant to the Reference time marker (PRN). While the pulse train of each satellite is very precise, there is nothing inherent with the atomic clock that guarantees the phase (leading edge of a pulse) of one atomic clock in one satellite will be in phase with another.

This is why the actual stream of pulses coming from each satellite is NOT the frequency of the atomic clock. Each pulse streaming from a satellite consists of an integral number of atomic clock cycles. Typically the integral number is 5 but is unique for the multiple pulse streams GPS satellites transmit (ie L1 stream, L2 stream, L5 stream).

With the pulse width coming from the satellite to the receiver being an integral multiple of the atomic clock, the manager of the GPS system is given the opportunity to shift the start of the pulses streaming from each satellite with the resolution of one atomic clock cycle and yet retaining the pulse width in each of these pulse streams to within one atomic clock cycle.

Extraction of information from pulse streams

Satellites emit pulse streams at different frequencies. Thus there is a pulse width associated with L1 specification, another with L2 specifications, etc.

What is important to grasp is that L1 from each satellite is the exact same frequency. So the interesting question is how does the receiver sort out which pulse belongs to which satellite.

That question is answered by the use of CDMA.