Figure 1a) One Satellite
To make GPS more understandable, the system is described through five conceptual pieces: satellite triangulation, distance to the satellites, perfect timing, determining where the satellites are in orbit, and ionospheric delays.
Triangulation from satellites is the basis of the system. To triangulate, GPS measures distance using the travel time of a radio message. To measure travel time, GPS needs very accurate clocks. Additionally, once the distance to a satellite is known, the position of the satellite in space must also be known. Finally, as the GPS signal travels through the ionosphere and the earth's atmosphere, it gets delayed.
Satellite Triangulation
Ignoring for the moment that we do know the distance from ourselves to the moving satellites, we can assume that we are 18,000 km away from satellite A, for example. This significantly narrows the scope of the search for our location. It tells us that we are on an imaginary sphere that is centred on the satellite having a radius of 18,000 km.
Figure 1a) One Satellite
It is also known that we are 19,000 km from satellite B. Presently with these two distances, we could be positioned anywhere on the circle where the two spheres intersect.
Figure 1b) Two Satellites
With a measurement from a third satellite say 20,000 km from satellite C, we can pinpoint ourselves if we disregard the ridiculous answer. As the three imaginary spheres intersect, only two locations are possible: one will be the correct answer, and the other will either be at an impossible altitude or an impossible velocity.
Figure 1c) Three Satellites
Mathematically we need four measurements to determine our location in three-space, but if we disregard the ridiculous answer we could theoretically proceed with only three measurements.
Distance to the Satellites
The GPS system works by timing how long it takes radio signals generated by the satellites to reach us, and then calculating distance from multiplying the speed of light by the time taken.
The problem with this is that we do not know when the signals left the satellites. To get around this, the system synchronizes the satellites and receivers so that they are generating the same codes at exactly the same time.
The digital codes called "pseudo-random" codes are generated by the satellites as well as the receivers. These carefully chosen codes repeat every millisecond.
Figure 2) Timing Difference
Perfect Timing
In the GPS satellites four atomic clocks are on board. One clock is needed for timing and synchronization, while the other three are for redundancy in case of a failure. Since atomic clocks are very expensive, the GPS receivers have clocks with nanosecond accuracy. Yet, even with this accuracy, the measurement can be off quite substantially.
In the previous section, it was said that theoretically three measurements were needed to get a position fix. However, with four imperfect measurements, for a 3-d fix, any timing offset can be eliminated, as long as all the offset are consistent.
Figure 3) Three Satellite Accurate Timing
As can be seen in the case of incorrect timing (Fig. 3), the spheres do not intersect at a single point. The onboard computer in the GPS receiver uses an algebraic algorithm to determine the timing error associated with the readings. Once this timing error is known the arcs will all intersect at one point, the current location of the receiver.
Determining Where the Satellites Are in Orbit
The satellites are in a non-geosynchronous orbit with an orbital period of 12 hours. During each orbital period the satellite passes over one of the Department of Defense (DoD) monitoring stations. The DoD monitors the satellites' altitude, position and speed. The variations they are looking for are called "ephemeris" errors: minor errors caused by gravitational pulls for the moon, sun, etc. This information is then relayed back up to the satellite. That satellite will then broadcast these minor corrections along with its timing information in a system "data message". This data message takes 30 seconds to read, and is required by the GPS receiver to accurately track the satellites' position.
Ionospheric Delays
The ionosphere is a blanket of electrically charged particles 129 km to 193 km above the earth. These particles slow down the signals coming from the satellites. The velocity of light through the ionosphere is inversely proportional to its frequency squared.1 To overcome this error, some GPS receivers use what is known as "dual-frequency" correction. This compares the arrival times of two different parts of the GPS signal which are at different frequencies and deduce the time delay associated with the ionosphere.
Frequencies of GPS
Carrier Frequency
Two carrier frequencies used by the GPS are 1227.60 MHz and 1575.42 MHz that are in the L-band frequency range.
Course/Acquisition (C/A) Code
The GPS code consists of 1023 binary pseudo-random codes at a chip rate of 1.023 MHz. A chip is the transition time for an individual bit in the pseudo-random sequence.
The C/A code is the code used by the NVI system to get positional information.
Pseudo-Random Codes
By using pseudo-random codes, GPS signals can be very low power and can still be picked up by an antenna a few centimetres across. The GPS receiver compares its internally generated pseudo-random code with the signals obtained by the antenna. As the number of matches starts to increase, the satellite's C/A code is found and can then be tracked and utilized to get positional information.
Pseudo-random codes are a sequence of 1023 binary bits that repeat every millisecond. These sequences are used to obtain timing information between the satellites and the GPS receivers.
Differential GPS and Pseudolites
The U.S. DoD purposely degrades the signals from the satellites for military reasons that introduce errors of + 100 meters to our readings. This is called Selective Availability or "S/A".
By placing a stationary GPS receiver at a known location, it can be used to figure out exactly which errors the satellite data contains. It acts like a static reference point. This receiver called a "Pseudolite" can transmit an error correcting message to any other GPS receivers that are in the local area. These additional receivers can use the error message to correct their positional solutions (Fig 4).
This concept works because the satellites are so far above the earth that errors measured by one receiver will be almost exactly the same for any other receiver in a given area. This correction factor not only corrects S/A but will also reduce ionospheric, timing, and atmospheric errors.
Figure 4) Differential Correction -- Pseudolite
