Global Positioning System

Technical description

The system consists of a "constellation" of at least 24 satellites in 6 orbital planes. The GPS satellites were manufactured by Rockwell; the first was launched in February, 1978 (Block I), and the final (24th), satellite was launched in 1994. Each satellite circles the Earth twice every day at an altitude of 20,200 kilometers (12,600 miles). The satellites carry atomic clocks and constantly broadcast the precise time according to their own clock, along with administrative information including the orbital elements of their own motion, as determined by a set of ground-based observatories.

The receiver does not need a precise clock, but does need to have a clock with good short-term stability and received signals from four satellites in order to find its own latitude, longitude, elevation, and the precise time. The receiver computes the distance to each of the four satellites by the difference between local time and the time the satellite signals were sent (this distance is called a pseudorange). It then decodes the satellites' locations from their radio signals and an internal database. The receiver should now be located at the intersection of four spheres, one around each satellite, with a radius equal to the time delay between the satellite and the receiver multiplied by the speed of the radio signals. The receiver does not have a very precise clock and thus cannot know the time delays. However, it can measure with high precision the differences between the times when the various messages were received. This yields 3 hyperboloids of revolution of two sheets, whose intersection point gives the precise location of the receiver. This is why at least four satellites are needed: fewer than 3 satellites yield 2 hyperboloids, whose intersection is a curve; it's impossible to know where the receiver is located along the curve without supplemental information, such as elevation. If elevation information is already known, only signals from three satellites are needed (the point is then defined as the intersection of two hyperboloids and an ellipsoid representing the Earth at this altitude).

When there are n > 4 satellites, the n-1 hyperboloids should, assuming a perfect model and measurements, intersect on a single point. In reality, the surfaces rarely intersect, because of various errors. The question of finding the point P can be reformulated into finding its three coordinates as well as n numbers r_i such that for all i, PS_i-r_i is close to zero, and the various r_i-r_j are close to C.Δ_ij where C is the speed of light and Δ_ij are the time differences between signals i and j. For instance, a least squares method may be used to find an optimal solution. In practice, GPS calculations are more complex (repeat measurements etc...).

There are several causes: The initial local time is a guess due to the relatively unprecise clock of the receiver, the radio signals move more slowly as they pass through the ionosphere, and the receiver may be moving. To counteract these variables, the receiver then applies an offset to the local time (and therefore to the spheres' radii) so that the spheres finally do intersect in one point. Once the receiver is roughly localized, most receivers mathematically correct for the ionospheric delay, which is least when the satellite is directly overhead and becomes greater toward the horizon, as more of the ionosphere is traversed by the satellite signal. Since it is common for the receiver to be moving, some receivers attempt to fit the spheres to a directed line segment.

The receiver contains a mathematical model to account for these influences, and the satellites also broadcast some related information which helps the receiver in estimating the correct speed of propagation. High-end receiver/antenna systems make use of both L1 and L2 frequencies to aid in the determination of atmospheric delays. Because certain delay sources, such as the ionosphere, affect the speed of radio waves based on their frequencies, dual frequency receivers can actually measure the effects on the signals.

In order to measure the time delay between satellite and receiver, the satellite sends a repeating 1,023 bit long pseudo random sequence; the receiver knows the seed of the sequence, constructs an identical sequence and shifts it until the two sequences match.

Different satellites use different sequences, which lets them all broadcast on the same frequencies while still allowing receivers to distinguish between satellites. This is an application of Code Division Multiple Access, CDMA.
There are two frequencies in use: 1575.42 MHz (referred to as L1), and 1227.60 MHz (L2). The L1 signal carries a publicly usable coarse-acquisition (C/A) code as well as an encrypted P(Y) code. The L2 signal usually carries only the P(Y) code. The keys required to directly use the P(Y) code are tightly controlled by the U.S. government and are generally provided only for military use. In spite of not having the P(Y) code encryption key, several high-end GPS receiver manufacturers have developed techniques for utilizing this signal to increase accuracy and remove error caused by the ionosphere.

A minor detail is that the atomic clocks on the satellites are set to GPS time, which is the number of seconds since midnight, January 5, 1980. It is ahead of UTC because it doesn't follow leap seconds. Receivers thus apply a clock correction factor, (which is periodically transmitted along with the other data), and optionally adjust for a local time zone in order to display the correct time. The clocks on the satellites are also affected by both special, and general relativity, which causes them to run at a slightly slower rate than do clocks on the Earth's surface. This amounts to a discrepancy of around 38 microseconds per day, which is corrected by electronics on each satellite. This offset is a dramatic proof of the theory of relativity in a real-world system, as it is exactly that predicted by the theory, within the limits of accuracy of measurement.

Continue to Sources of GPS measurement errors.

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