Copyrights 1997, 2002
Chapter 2, GPS operations
Now, let’s get a little technical. We said in Chapter 1 that GPS ranging codes from the satellites and from the receivers must be adjusted so that they are synchronized with each other. In the Holmes positioning system the radio broadcasting station handles the synchronization for Big Ben and the radio receiver, but there is no such mechanism for the GPS satellites and your GPS receiver. So how can we be sure that that all the satellites and receivers are perfectly synchronized, as we assumed in the previous chapter that they were?
If our system is out of sync by just one millisecond—1/1,000thof a second—then our range (distance) measurement will be off by 186 miles (300 kilometers), since radio waves travel at 186,000 miles (300,000 kilometers) per second. This kind of error is intolerable! Well, the satellites have atomic clocks on board. These clocks are extremely precise and very expensive; they cost a hundred thousand US dollars each, and each satellite has several of them just to make sure that at least one will be working. And your GPS receiver, down here on the earth? It has a quartz clock, just like the one in your wrist watch, and quartz clocks are not as precise as we need them to be.
Fortunately, we can use an extra range measurement to make up for the imperfect sync of your receiver. In three-dimensional space there are three unknown coordinates to be determined; i.e. x, y, and z. Our high school algebra books tell us that we need three independent equations to solve three unknowns. We can use three range measurements from three satellites to set up the three equations we need. But we also have an extra unknown—the clock error or clock bias of your receiver, which we represent by the notation terror. Because of this fourth unknown, we need a range measurement from a fourth satellite, and a fourth equation. So we need four equations to solve four unknowns: x, y, z, and terror. (See Fig. 2-1)
Solving these equations is not an easy job for us human beings; fortunately, however, the microcomputer and the software that is running on it, which we mentioned earlier, can handle the mathematics easily.
Fig. 2-1 Four satellites are needed to determine exact position and time.
This is how we get the position (x, y, and z) and the time precisely. Getting the exact time from this system is a GPS bonus, and there are many applications in telecommunications and other fields where this precise time is needed—for example, in the control of data communication networks, frequency hopping communication, and TDMA systems, where precise time permits all users to synchronize with each other. There are even GPS receivers that are dedicated only to the extraction of the precise time.
They don’t have to stay there
Our analogy of Holmes’s Big Ben positioning system to GPS may mislead you into thinking that GPS satellites are in geo-stationary orbit, because Big Ben is located at a fixed place. (A geo-stationary orbit is also called a geo-synchronous orbit. Satellites in geo-stationary orbits circle the earth once every 24 hours and travel in the same direction as the earth’s rotation, so from the perspective of someone on the earth’s surface they remain fixed at the same location in the sky.)
In fact, though, the source of a signal does not need to be stationary like Big Ben, if somehow we can predict its location and motion at all times. Fortunately, the satellites orbit the earth in a very predictable way. The theoretical problem for predicting their location and motion was actually solved a long time ago by the great German astronomer and mathematician Johannes Kepler (1571-1639). He discovered the laws and equations needed to describe the movements of heavenly bodies—the moon around the earth, planets around the sun, etc. Man-made satellites also obey Kepler’s laws.
Of course, the monumental achievements of Sir Isaac Newton (1642-1727) and Albert Einstein (1879-1955) have also contributed to the GPS story. We will talk about those contributions in later chapters. There are, indeed, complicated mathematical problems to be solved in calculating you position using the GPS; but again, the powerful microcomputer insider your receiver, running millions of instructions per second (MIPS), uses the programmed mathematical formulas of Kepler's law to solve the problems and compute the locations of the satellites--and of your receiver--almost instantaneously.
Fig. 2-2 These men and other scientists contributed to the theoretical foundations of
What is terribly difficult for a human being to compute is a piece of cake for that modern wonder, the computer, and its software engineering, because the computer works at the speed limit of the Universe--that is, the speed of light. No matter how complicated a computing process is, the computer can do it in just a fraction of a second.
The orbits of the GPS satellites are known, and they broadcast a rough set of orbital information known as the almanac which tells you where to look for each satellite in the sky at any given moment. Your receiver reads this almanac and writes the information into its non-volatile memory (NVM), so that it won't forget when you turn the device off. So we now see that the GPS satellites not only transmit the ranging pseudo-random codes, but also send "data messages" about their orbital locations and other things.
Why not stay in place?
But, you may wonder, wouldn't it be simpler to use geo-stationary satellites? There are several reasons why this isn't done. First of all, geo-stationary orbits have to be right over the equator, and users near the poles have very poor visibility of satellites in those orbits. (See Fig. 2-3)
Fig. 2-3 Satellites in geo-stationary orbit travel 36,000 kilometers above the equator and complete one circuit every 24 hours, making them synchronous with the earth's rotation so that they stay above the same spot on earth all the time. Many communications satellites use this orbit, but it is not suitable for GPS satellites because they have to be spread out all around the world--not just above the equator.
Fig. 2-4 The GPS Constellation consists of 24 satellites (three of which are spares). They travel 20,200 kilometers above the earth's surface in six different orbital planes, so that they can cover the globe evenly. Their orbital period is approximately 12 hours. Four or more of the satellites are always "visible" from any location on earth at all times. (From "GPS NavStar User's Overview," by ARINC)
If all of the satellites were in the same plane, as in Fig. 2-3, there would be a very high Dilution of Precision (DOP) and very poor results--perhaps even an inability to make to good position fix.
In fact, as you can see in Fig. 2-4, the GPS satellite orbits are evenly distributed all around the globe. These satellites circle the earth two times a day, with an orbital period of approximately 12 hours. This satellite constellation needs to be monitored and communicated with by a number of ground control stations operated by the U.S. Department of Defense (DOD). For the ground stations, being able to "see" the satellites twice a day is a convenience that enables them to check the satellites' positions and "health," and to upload data messages for broadcasting to users like you.
The ground stations need to measure and predict the satellites' position and speed precisely, because variations are caused in the ideal Keplerian orbits by things like the solar wind and the gravitational pull of the moon and sun. A set of complete and unambiguous descriptions of a satellite's orbit and clock parameters is called an ephemeris. The satellites broadcast this ephemeris data to your receiver, which processes them and uses them to make precise position fixes.
If the control segment finds that a satellite has deviated from its predicted orbit, it corrects the deviation by uploading new ephemeris message data to the satellite for broadcasting. The correction is not usually made by maneuvering the satellites back to their precise predicted orbit, because the variations are very small and it would be too difficult for the control stations to do that kind of thing all the time, especially since the fuel available for maneuvering is very limited. The satellites are physically maneuvered only occasionally, when the control sector finds it difficult to predict ephemeris parameters for them.
GPS as a whole
You should have a clear picture of the GPS system by now. The GPS community is made up of three segments: there is the User Segment, which includes you; there is the Space Segment, made up of the satellites; and there is the Control Segment, comprised of the ground control and monitoring stations. (See Fig. 2-5)
Fig. 2-5 Three segments of the GPS system
(From "GPS NavStar User's Overview," by ARINC)
Four satellites travel in each of six different orbital planes inclined about 55 degrees to the equator, at an altitude of 20,190 kilometers above the earth. There is a total of 24 of these satellites.
The control segment consists of five monitoring stations, four ground antenna upload stations, and an Operation Control Center. The site of each of these ground facilities has been carefully chosen to provide a proper longitudinal separation between them.
Since the GPS satellites are not in geo-stationary orbit, there are relative speeds between you and the satellites. This causes the Doppler effect. (See Fig. 2-6)
Cartoon drawn by Jyh Hong Hsieh of the Center for Aviation and Space Technology
Fig. 2-6 A plot of the Doppler frequency change as the source moves toward the
receiver, passes it, and them moves away from it.
You may be familiar with how the pitch of an ambulance or police car siren changes as the vehicle approaches, passes by, and then moves away. The sound starts out with a high pitch as the source comes toward you, drops noticeably as the vehicle passes by, and then stays at a lower pitch as it recedes into the distance.
The change that you heard in the siren's pitch is called the "Doppler shift" of the sound waves. It is caused by the relative movement between you and the ambulance or police car.
Another Doppler phenomenon is useful for soldiers in combat, but let's hope we never experience it in our everyday lives. Soldiers under artillery bombardment learn to tell which shells are coming directly toward them, so that they should take cover immediately, by the sharp howling sound they hear moments before the shells hit. But shells flying overhead toward some distant landing point create sound of a lower pitch, so the soldiers learn to ignore them.
Radio and light-wave frequencies also shift if the source (transmitter) and observer (receiver) are in relative motion. The microwave signals that GPS satellites send out also shift in frequency because of the relative motion between your receiver and the satellites.
This change in frequency--the Doppler shift--is independent of the ranging code time delay that we discussed in the previous chapter. It can be detected very easily by your receiver, and is very useful in determining your speed and the direction you are moving in. This is another advantage that GPS satellites have over those in geo-stationary orbit.
The primary functions of a GPS receiver are to determine your "PVT"--Position, Time, and Velocity. (We have mentioned the first two of these measurements previously.) Some GPS users may intuitively assume that speed is determined simply by differentiating the position--that is, by dividing the position change by the time interval. But this is not the way a good GPS receiver does the job. Position measurements are noisy, uncertain, or random in nature, and differentiation of such a noisy quantity is not a good practice. Instead, velocity or speed should be determined by using the Doppler shift of the GPS satellite signals.
Notice that if a satellite is directly above the receiver, the satellite's movement is perpendicular to the line of sight (from receiver to satellite) and there is no Doppler shift between the two at that particular moment. (See Fig. 4-1 in Chapter 4) To measure velocity to centimeter-per-second accuracy, centimeter-wavelength signals are needed. The wavelength of a microwave is in the centimeter range, and microwaves are what GPS uses.
An earlier-generation satellite position deployed by the U.S. Navy, the TRANSIT system, uses Doppler shift measurements alone to determine the user's position. This system is still operational. But since the movement of the receiver changes the Doppler shift and thus introduces error, this system is very sensitive to user movement and cannot be used for aircraft navigation.
We all know that the Hubble space telescope, which was deployed by the space shuttle, has made exciting new discoveries about our universe recently. This telescope was named after the American astronomer Edwin Powell Hubble (1889-1953), who made Doppler observations and formulated a law to describe the relationship between distance and the Doppler shift. Doppler effect is a very useful measurement in many applications.
Radar and sonar
Now let's say a few words about other ranging technologies we've all heard about, radar and sonar. These have been around for some time, and they have played a significant role in modern history ever since their early development. It has been said that these technologies saved England (and, fortunately, helped keep our Big Ben intact) during the Battle of Britain. Radar helped achieve effective command and tactics in the air, while sonar helped break the submarine blockade at sea. This made it impossible for the Luftwaffe (the German Air Force) to control the skies, and for the "wolf packs" of U-boats (German submarines) to choke off the sea lanes that kept England supplied. This is why the Germans could not invade the British Isles and the Allies could return to the European Continent in World War II.
Radar and sonar are still serving the needs of defense, air traffic control and safety, and ocean resources research all over the world, even today.
(RAdio Detection And Ranging) and SONAR (SOund Navigation And Ranging) also
employ extensive Doppler processing for the detection and speed measurement
of moving targets and for the creation of radar and Sonar imagery. They send
out phase coherent pulses of electromagnetic or sound waves and detect back-scattered
waves (returns) from the target. Stationary targets (clutter) are always
the same number of wavelengths from the re-transmitter and returns do not change
phase from pulse to pulse and may be filtered out. Waves from moving targets
will have a different phase from pulse to pulse because the range is changing.
This pulse to pulse change is detected and serves to separate the signal from
the clutter. These systems also employ sophisticated phase coding and compression
techniques in order to reduce the peak transmitted power requirements and to
improve the ability to resolve targets in range. Because radar and sonar energy
must travel in a round trip from the transmitter to the target and back, the
distance to a target is determined by using half of the elapsed time from the
transmission of the pulse to its return. The range is R = ct/2 where c is
the speed of sound or light, as the case may be, and t is the elapsed time
for the round trip.
GPS ranging, on the other hand, is a kind of one-way communication and ranging technology (with satellites broadcasting the data messages and sending the ranging codes in only one direction, as we explained previously). This is an extremely low-powered system—so low-powered, in fact, that the signal is really buried in the natural background radio noise. The receiver recovers this weak signal from the satellites by using spread-spectrum communication technology, a very tricky concept in communication theory that works by means of pseudo-random codes. Spread-spectrum communication is also used by spacecraft that are sent to explore other planets in the solar system, where distances are so great that that the signal coming from the craft in deep space is so attenuated that it too is buried in the background noise by the time it reaches the earth. Mathematically, the power or strength of a signal is inversely proportional to the square of the distance it has traveled. These signals are a form of electromagnetic energy which, like gravitational force, luminescence, and other things, is the nature of our 3-dimensional space.
There has to be a way to recover and amplify this weak signal, so spread-spectrum communication with pseudo-random code technology performs a sort of “amplifying effect” on the signal so that it can be recovered by the receiver. This is another advantage of the pseudo-random code, so now we have learned about a third function of this kind of code in the GPS system. The other two, which we explained in Chapter 1, are determination of delay time and identification of satellites.
Spread-spectrum communication appears to amplify itself without requiring the input of huge amounts of power, so you don’t need a large and expensive dish antenna aimed at a particular satellite to focus the signal, like a satellite TV receiver does. The antennae used by GPS receivers are not as efficient as satellite TV dish antennae; but they are very small and handy, since they don’t have to be aimed at any particular GPS satellite. They are called omni-directional antennae. Imagine what a nightmare it would be if your GPS antenna had to be aimed at several moving GPS satellites that you can’t even see!
A free lunch?
This might make you wonder: If this spread-spectrum technology is so good, why don’t TV satellites use it? Well, as the old saying goes, “there is no such thing as a free lunch.” Application of the pseudo-random code principle requires comparison of signals over a relatively long period of time, making it a slow and cumbersome process. A lot of time is needed to sort out the extremely weak GPS signals from all the background noise that GPS receivers pick up. This means that the transmission of data messages from GPS satellites can be very slow*—only 50 bits per second—so that it takes 12.5 minutes for a GPS satellite to send out its full set of data messages. This kind of communication is too slow for a TV signal, which is loaded with image information and needs a very high transmission rate or bandwidth to carry it. In other words, GPS uses spread-spectrum pseudo-random code communication technology in such a way that data transmission speed is sacrificed in exchange for low-power performance and omni-directional antenna characteristics. This, of course, is a practical trade-off.
*Spread-spectrum communication alone does not make the transmission very slow. GPS system design considerations have the GPS data rate as 50 bits/sec.
Chapter 3, GPS mathematics and physics
In previous chapters we explained how Sherlock Holmes III determined distance by listening to bell chimes and observing their pattern. Your GPS receiver measures distance by detecting extremely weak GPS satellite signals with its terribly complicated electronic circuits—its hardware—and analyzing those signals with its amazingly intricate software.
So perhaps you are thinking that the cost of a GPS receiver must be awfully high. And indeed it was forbiddingly expensive for us ordinary consumers—back in the 1980s, in the early stages of the GPS’s development and deployment. But no more.
The Industry Trend
Today, at the turn of the millennium, we have much cheaper and much better GPS receivers. They perform more reliably, and they do more things. Even better, their cost is continuing to fall year by year and we’re certain that it will keep declining in the foreseeable future. How can such a sophisticated device be so inexpensive that its cost compares to that of other consumer products?
Thanks to the advances in semiconductor technology and microelectronics design automation that have taken place over the past two or three decades, the complicated electronic circuits used in GPS receivers (and other electronic products) can now be integrated into a small silicon chip about the size of the nail of your little finger.
These miniaturized circuits on single chips are called integrated circuits, or ICs. If an IC is designed for a specific purpose like the processing of GPS signals, it becomes a custom-made integrated circuit called an “application specific IC,” or ASIC. If an ASIC can be made in large quantities, the cost of each individual unit will be very low. This is why we can afford to enjoy a lot of “intelligent” consumer electronic products at home. So, as long as the population of GPS users keeps growing, we can be sure that the cost of GPS receivers will keep going down.
It may surprise you to learn that the software used in a GPS receiver can cost many hundred man-years of joint efforts to develop, or write. But this is not a cost that recurs, like it does in the manufacturing of hardware. Once the software is written and debugged, there is no more cost no matter how many receivers are made. For this reason, the high cost of developing the software can be shared among the huge number of users.
Much of the recent software has been written in C language. This is a popular programming language that has been taught in many schools and colleges since the mid-1980s.
In previous chapters we also described some of the details of the basic principles behind GPS. We noted that GPS uses satellites as reference points for triangulating your position on the earth, in the air above the earth, or even in space. Since we know right where all of the satellites are in their orbits, we can use their positions to triangulate ours.
Triangulation and the Cartesian coordinate systems used by GPS for the computing of positions are based on the Pythagorean theorem, which describes the relationship between the lengths of the three sides of right triangles. This relationship is
a2+b2=c2. See Figure 3-1.
Fig. 3-1 The Pythagorean theorem
Precision and Precession
Since the GPS is designed to have the potential of achieving a precision of location to less than an inch, the message data from the satellites and the receivers must be able to correct all sources of errors, no matter how small they are. We will talk about some of that in this chapter, and we will also discuss the earth’s imperfect rotation, the influence of atmospheric effects, and GPS time.
Fig. 3-2 Precession of the earth’s rotation. The axis around which the earth rotates wobbles somewhat, sweeping out a circle in space. Since it takes 26,000 years for the axis to sweep out a full circle in space, the best gyroscope on earth is the earth itself.
Sir Isaac Newton, who discovered the law of gravity, also knew about the precession of the earth’s rotation. Just like a spinning top that is subjected to a force (or torque, to use the precise term), the axis of the earth’s rotation tilts and changes direction, thereby describing a circle in space. The precession period of the earth is 26,000 years, making it very slow indeed. It causes the vernal equinox (see Appendix A) to move westward very slowly, at a rate of 50.27 seconds of arc every year or 1.39697128 degrees of arc per century. (See Fig. 3-2)
Note: Figure 3-2 here is planned to animate Earth's precession of the rotation, behaving like a tilted rotating top. However, we don't have the web site expertise to design such a illustration yet. If you know how to do that please let us know by mailing to: firstname.lastname@example.org .
During the more than 2,000 years since the Greek astronomer Hipparchus named the twelve constellations of the zodiac after figures from Greek mythology and legend, precession has caused the vernal equinox to move from the constellation of Aries (the Ram) to Pisces (the Fish).
Anyone who had discovered this fact alone would have been remembered for all time as a great physicist. It is amazing and amusing that for Isaac Newton, who described, explained, and calculated the drift of the equinox caused by the precession of the earth, which up until his time had been an astronomical mystery, this achievement is lost in the wealth of his accomplishments. We often fail to grasp how difficult, and how significant, this task was at that time in history.
Precession and other small irregular variations in the earth’s rotation affect the orbital positions of the GPS satellites, and the corrections for those variations are broadcast in the satellite data messages. The software program inside your receiver uses this data to fine-tune its distance computations.
Effects of the atmosphere on GPS
At the beginning of this book we mentioned how radio waves, microwaves, visible light, X-rays, and other waves all travel at the speed of light, because they are all different forms of electromagnetic waves. While they are all the same kind of waves, they have different frequencies. James Clerk Maxwell (1831-1879), a Scottish physicist, asserted this fact as a result of his pure theoretical studies. The wave spectrum is illustrated in Fig. 3-3.
So far, we have assumed that the radio signals transmitted by the satellites travel at the constant speed of light—to be exact, c = 2.99792458X10 8 meters/second. This is the speed specified in the GPS system, and it is the most precise value ever available. But this is the speed that is achieved in a vacuum, as in space. Just as sunlight is refracted through a prism because the different color wavelengths which it includes travel at different speeds in glass (see Fig. 3-4), a satellite’s signals are also affected when they enter the different parts of the earth’s atmosphere.
Fig. 3-4 The colors contained in sunlight are dispersed by a prism because their
different frequencies slow down differently in glass.
Newton, again, discovered the dispersion of colors in sunlight. Notice that the color red is refracted less than violet. This is because red light has the lowest frequency and violet the highest frequency in the spectrum of visible light, and the lower-frequency signal is refracted less than the higher-frequency signal. This phenomenon is explained by the fact that waves of different frequencies slow down by different amounts as they pass through the prism glass.
In much the same way, our GPS signal is slowed down and bent when it enters the earth’s atmosphere. This atmospheric error can be divided into two types, ionospheric and tropospheric. The primary error occurs in the ionosphere, a layer of electrically charged particles that reaches from 80 miles to 200 miles above the earth. The density of the charged particles in the ionosphere varies from day to night and also from season to season, depending on the amount of the sun’s energy that reaches the atmosphere. Since the delay of the radio signal depends on the density of the charged particles in a way that is somewhat predictable, some civilian GPS receivers use a mathematical model of the delay, which is broadcast by the satellites, to reduce the error.
A better way to measure the radio signal’s delay is by comparing two signals of different frequencies. This is just like the color dispersion we considered above, where the lower frequency of red is refracted less than the higher frequency of violet because the lower frequency gets delayed (or slowed down) more than the higher frequency. There is a formula to describe this phenomenon: a signal slows down at a rate inversely proportional to the square of its frequency. This is the method of determining ionospheric error that is usually used in military equipment and in high-end, expensive, dual-frequency GPS receivers.
GPS satellites broadcast on two frequencies: L1, or 1575.42 megahertz, and L2, or 1227.6 megahertz. Dual-frequency receivers can receive both of these frequencies and carry out the ionospheric correction by making measurements simultaneously on these two well-spaced frequencies. This calculation can remove most of the ionospheric error. Most civilian receivers, however, can receive only the L1 frequency, because the cost of dual-frequency receivers is too high.
The troposphere, which reaches upward from the earth’s surface, is where all of our weather is. Water vapor in the troposphere can affect radio signals, and the delay it causes them can also be approximated by mathematical models. The software algorithm programmed into your GPS receiver uses these mathematical models to make adjustments for the effects of the atmosphere on its position computation.
This might be a good place to introduce a common engineering practice to you. In the last section, we mentioned the use of mathematical models to correct the effects of the atmosphere on GPS receivers. Many times, engineers use mathematical models to solve problems. One example of this is the use of empirical formulae. Automobile engineers often employ such a formula to describe the behaviors or characteristics of an engine in order to control them effectively. These formulae, or equations, are not formulated on a theoretical basis; instead, they are derived from experiments or observations. An equation like this can be very complicated, since there are so many factors that affect the performance of an engine. In GPS applications, yet another example of empirical formulae is the use of the ephemeris. This is essential, because the equations used by GPS receivers to correct for the solar wind, the gravitation force of the moon, and other such factors on the GPS satellites are too complicated to be predicted by theoretical means. They are not like Kepler's equations, which can be deduced from Newton's theory of universal gravitation. So the only practical way to solve these real-world problems is, again, by using mathematical models.
Your GPS receiver also has another way to utilize math models. It uses an ellipsoidal model of the earth's shape to estimate the latitude, longitude, and altitude of the receiver. The equatorial radius of the earth is 6,378.137 kilometers and its polar radius is 6,356.762 kilometers; the difference is due to a slight flattening of the earth caused by its spinning. Because it does not coincide precisely with the earth's surface, this model causes discrepancies in the altitude calculation of the GPS receiver; it is necessary, however, because there is no better way to represent the real world. We will talk a little about this in Chapter 5, in the GPS and GIS section.
We noted earlier that precision satellite clocks and time measurements are the key to the accuracy of GPS positioning. GPS time is based on atomic standard time, and is continuous; there are no leap seconds like there are in UTC, or universal coordinated time, because the global positioning system cannot tolerate any such discontinuity in the time count. The GPS navigational messages that the satellites broadcast provide correction parameters for GPS time.
We will talk more about time and times in the next chapter.
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