The
system consists of a "constellation"
of at least 24 satellites in 6 orbital planes.
The GPS satellites were initially manufactured
by Rockwell; the first was launched in February,
1978, and the most recent was launched on November
6, 2004. Each satellite circles the Earth twice
every day at an altitude of 20,200 kilometres
(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 receive 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
from 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 4 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 imprecise 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, or CDMA.
Several
frequencies make up the GPS electromagnetic
spectrum:
-
L1 (1575.42MHz):
Carries a publicly usable coarse-acquisition
(C/A) code as well as an encrypted P(Y)
code.
-
L2 (1227.60MHz):
Usually carries only the P(Y) code. The
encryption 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. The keys are changed
on a daily basis. In spite of not having
the P(Y) code encryption key, several high-end
GPS receiver manufacturers have developed
techniques for utilizing this signal (in
a round-about manner) to increase accuracy
and remove error caused by the ionosphere.
-
L3 (1381.05MHz):
Carries the signal for the GPS constellation's
alternate role of detecting missile/rocket
launches (supplementing Defense Support
Program satellites), nuclear detonations,
and other high-energy infrared events.
-
L4 (1841.40MHz):
Being studied for additional ionospheric
correction.
-
L5 (1176.45MHz):
Proposed for use as a civilian safety-of-life
signal.
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 6,
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. |
