Numinations — November, 1998

Global Positioning

© 1998, by Gary D. Campbell

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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