No matter where you go, these days it’s possible to know exactly where
you are. Armed with a ten ounce GPS receiver and a map—one with
latitude and longitude markings—you can locate yourself within a
hundred feet or so and navigate perfectly. All this with no special
skills or knowledge? But, how, you might ask?

We’ve had maps for thirty years or more that were up to this task.
Many areas of the world are covered by excellent maps. But, even with
a good map of the area you are in, the problem has always been to know
exactly where you are on the map, and which direction on the map
corresponds to the direction you are heading in the real world. Now
(and for the past couple of years), anyone can know these things by
spending about $150.

So, what is a GPS receiver, and how does it work? A GPS receiver picks
up the radio signals from a constellation of Global Positioning
Satellites. Each satellite orbits the earth about once every twelve
hours. The Department of Defense has anywhere from twenty-four to
thirty-two of these satellites in the sky at any given time. They are
in semi-polar (or at least non-equatorial) orbits. They orbit about
11,000 miles overhead. Each one has a mass of about one ton and is
about 17 feet across with its solar panels extended. Each one carries
at least one computer, atomic clock, and radio transmitter. Most have
working spares of all these devices aboard. Each satellite also has a
receiver that listens only to the Department of Defense to tell it what
to do. This gives the satellites periodic updates so that each
satellite knows, within a few feet, where it is at any given
nanosecond. However, the good old DoD can also direct the satellites
to report their positions inaccurately. This keeps enemies (and, of
course, us civilians) from using the system for our own nefarious
purposes, such as sending over an ICBM with pinpoint accuracy, or
avoiding collision with an underwater obstacle in a harbor.

A GPS receiver needs to acquire the signals of at least three, but
preferably four, satellites. Given that there are at least twenty-four
satellites in the sky, about half of them can be seen from any point on
the ground; the other half are on the other side of the world. Maybe a
third to a sixth of them will be high enough in the sky to provide a
decent signal. This generally gives you four usable satellites at any
given time and place.

Now that you have an overview of the system, there are three topics
that will enable you to know how the whole thing works: What
information is broadcast by the satellites? How do the receivers work?
What geometry is involved?

Let’s take the geometry first. Given that there are three or four
satellites overhead, each about eleven thousand miles away, suppose we
knew within a few feet exactly how far they were away from us and
exactly where they all were located? If we knew all these things, we
could figure out exactly where we were.

Let’s start with the first satellite. All the points equi-distant from
this satellite describe a huge sphere around it. We only care about a
point on this sphere that just touches our own position. Now, take the
second satellite. Again, we consider the point on the sphere around
that satellite that just touches our position. Two spheres make a
circle when they intersect. Our position is a point somewhere on this
circle. Now, add the information from a third satellite. It defines
another sphere that just touches our location. A sphere intersects a
circle at two points. In this case, one of the points will generally
be nearer the surface of the earth than the other. Since our location
is probably at a point between sea level and a few thousand feet, that
makes one of the points an obvious choice. So, with only three
satellites, the GPS receiver can make a good guess as to where we are.
In fact, if the clock in the receiver is perfectly synchronized with
the clocks in the satellites (which are kept in near perfect time with
each other), it could have narrowed our location down to the regions
around two points with just the information from three satellites.
However, GPS receivers don’t keep perfect time. Not for $150. So, you
need the signal from a fourth satellite to resolve your location and
altitude, and to synchronize the clock in your receiver.

Now, in a general way, we know what must be broadcast from the
satellites and what the receivers do with it, but let’s get a little
more specific. All the satellites broadcast a digital signal on the
same frequency and at low power. Each signal being broadcast contains
three types of information: Timing information, identification
information, and a complete package of data about that satellite.
Because the signals are so weak and all of them are broadcast on the
same frequency, the first order of business for the GPS receiver is to
“tune” into a known satellite. This is like trying to listen to a
particular conversation in a crowded room with dozens of conversations
going on at once. You can’t do it. But, suppose your name is spoken?
Now, your attention is drawn to that conversation. If your name keeps
cropping up in one conversation, you will probably have no difficulty
“tuning it in.” This is how the receiver sorts out the babble coming
from all the satellites at once, and from other sources of noise. It
knows each satellite’s “name” and each satellite repeats its name
constantly.

This is quite ingenious; it’s worth going into further. The digital
signal from each satellite is broadcast at a little over a gigahertz (a
billion cycles every second). The “name” of each satellite is a unique
string of 1023 bits. Each bit takes about one microsecond, or about
1000 cycles of the carrier frequency, to be broadcast—about a
millisecond for the entire “name.”

It’s by knowing in advance the exact content and construction of each
satellite’s “name” that a receiver can lock onto a satellite’s signal.
This enables it to discriminate between the signal from that satellite
and the background noise, which includes the signals from all the other
satellites. Thus, the receiver can identify the strongest satellite
signals in its vicinity and “listen” to these several conversations,
one at a time, rotating among them.

Once it has identified three or four satellites, a receiver begins to
gather data from them. It can take the receiver several minutes for it
to collect all the data it needs. These data include an update on each
satellite’s position, its orbit, and the exact universal time at which
that satellite “speaks” its own name. Once these facts have been
registered, the receiver can use them in conjunction with the geometry
of the situation and draw conclusions about its exact location every
second or so.

After signal acquisition, and after data collection, the receiver
enters the mode of continuous update of its location. This mode
requires exact timing. Think about the string of 1023 bits that is the
“name” of each satellite. Each bit takes about one millionth of a
second to broadcast, and consists of about a thousand cycles of the
carrier frequency. Once the receiver has acquired the exact universal
time and knows the exact time that a particular “name” was broadcast,
it can pinpoint the time that a particular cycle of the carrier wave
was emitted by the satellite that broadcast it. The signal travels at
the speed of light. The speed of light is about a billion feet per
second, or one foot in a billionth of a second. This is the time it
takes for each cycle of the broadcast, so if you can pinpoint the time
a particular wave was emitted from a particular satellite, you can
pinpoint within about one or two feet how far you are away from that
satellite.

But, it’s not quite that perfect. The clocks can be off by a
nanosecond or two, so this adds about two feet of uncertainty to your
position. The position of each satellite is only known within a couple
of feet. The receiver adds another four feet, or so, to the error.
Atmospheric conditions add about twelve feet. And the good old DoD
adds about 25 feet of its own. Under most conditions, the geometry
also dilutes the precision by a factor of 4 to 6, so the total error
can be as much as 270 feet, or (2 + 2 + 4 + 12 + 25) x 6. However, by
averaging successive computations, a good receiver can usually narrow
this down to around a hundred feet on your map, and give your altitude
within a couple of hundred feet as well.

If the receiver is moving, it computes how fast and in which direction
it is moving. It can act as a “perfectly” accurate clock, a compass, a
speedometer, an altimeter, a what-have-you. In fact, a good GPS
receiver knows almost everything about where you are except things like
the sports and weather being played out around you. It can even
compute your local time of sunrise and sunset.

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