Chapter 2Introduction to the Global Positioning SystemIn Chap. 1 we introduced some of the working principles of the GPS by analogywith a simple linear model. Now its time to move into more detailed discussion ofthe GPS satellite system and how it works. In this chapter we will cover the basicsof how the satellites are conﬁgured, making the path delay measurements, a quicklook at the user position solution equations, and a high-level block diagram of GPSreceiver will be presented.2.1 The Satellite SystemGPS is comprised of 24 satellites orbiting the earth at a distance of 20,000 km asmeasured from mean sea level. See Fig. 2.1. Figure 2.2 shows a more detailedpicture of a user getting position information from four satellites. Each satellitecontains a very precise clock. All the clocks and hence all the timing signalsassociated with each satellite are in near-perfect synchronism. The basic principleof operation is similar to the Roadway Distance Measurement example of Chap. 1.A rough model is to imagine a system of satellites emitting a “pulse” of light atexactly the same instant as all the others (Important: GPS is NOT a pulsed system).If you were at the exact center of the earth and the satellites were in perfect circularorbits (GPS orbits are nearly circular) these pulses would all arrive at your receiverat the exactly same instant. For all other points the pulses would (typically) arrive atdifferent times. If one has knowledge of the positions of the satellites and canmeasure the path delay (distance) from at least four satellites to the user receiver,the user position X, Y, Z in ECEF (Earth Centered Earth Fixed) coordinates and userclock bias can be solved for.D. Doberstein, Fundamentals of GPS Receivers: A Hardware Approach, 23DOI 10.1007/978-1-4614-0409-5_2, # Springer Science+Business Media, LLC 2012
24 2 Introduction to the Global Positioning System c DKD INSTRUMENTS EARTHc DKD INSTRUMENTSFig. 2.1 The earth and its constellation of GPS satellites2.2 Physical Constants of a GPS Satellite Orbit That Passes Directly OverheadFigure 2.3 shows a simple model of one satellite circling earth directly overhead atthe user zenith. We will use this simple model to compute some constantsassociated with a user at or near the earth’s surface. We will assume that the usercan be at a maximum altitude above sea level of 10 km. The minimum altitude willbe assumed to be sea level. With this information and the known mean diameter of aGPS orbit we can calculate the minimum and maximum distance to the satellite as itpasses overhead. Figure 2.3 shows the elements used in this calculation and otherresults. From this information we ﬁnd that the maximum distance a GPS satellite is25,593 km. The minimum distance is found to be 20,000 km. These distancescorrespond to path delays of 66 and 85 ms, respectively. As we have seen inChap. 1, we can use this knowledge of maximum and minimum distance to setthe receiver’s reference clock to approximate GPS time.
2.3 A Model for the GPS SV Clock System 25 ZK R3 R1 EARTH R4 c DKD INSTRUMENTS USER EQU ATOR IA L PL R2 ANE YKXK c DKD INSTRUMENTSFig. 2.2 User at earths surface using four SV’s for position determination2.3 A Model for the GPS SV Clock SystemEach GPS SV has its own clock that “free runs” with respect to the other clocks in thesystem. GPS uses a “master clock” method in which all the clocks are “referenced” tothe master clock by the use of error terms for each SV clock. We discussed thisalready in Chap. 1. In this chapter we will use the clock model of Fig. 1.7 minus theoutside second-counter dial. In addition to the omitted second-counter dial, we willnot show the dial that indicates the week of the year, year dial, etc. We have notdiscussed these new dials but they are present in the time-keeping method employedby GPS. For the purpose of discussion and understanding GPS we will often use theclock model of Fig. 1.7 minus the dials above the 0–1 s time increment. If we show ordiscuss a SV or Receiver replica clock without all the dials, the reader will assume themissing dials are “present” but not shown for reasons of clarity. The smallest time increment dial of our clock model will always be the0–0.977 ms dial. As mentioned in Chap. 1 this dial has a very ﬁne “effective”resolution as used in the SV. But this statement is an approximation. In reality the0–0.977 ms dial is the one dial of all the clock dials in our model that does not have adirect physical counter part in the “true” SV clock. In other words, our clock modelat the sub microsecond level is not a completely accurate model of the GPS clock.
26 2 Introduction to the Global Positioning System SATELLITE ORBIT RMIN RMAX USER USER HORIZON LINE Rsv EARTH RADIUS OF EARTH ~ 6,368 KM c DKD INSTRUMENTSUSER TO SATELLITE DATA GPS SATELITE DATA R ~ 20,000KM MIN MEAN ALTITUDE ABOVE SEA LEVEL ~ 20,000KM ~RMAX 25,593 KM Rsv = RADIUS OF ORBIT WRT CENTER OF THE EARTH ~ 26,550 KM ORBITAL RATE : 2 ORBITS IN 24 HOURSMin Time Delay ~ 66msec ORBITAL SPEED ~ 3874M/SECMax Time Delay ~ 86msecFig. 2.3 The range from the receiver to a GPS satellite and physical constants (for a GPS satelliteorbit that passes directly overhead)When we use the clock model to describe the receiver’s reference clock (or SVreplica clocks) we will see that the 0–0.977 ms dial does have a direct physicalcounter part in the receiver. The impact of having the smallest time increment dialnot an exact model for the SV clock is not an important issue for this text as our goalis position accuracy of Æ100 m.2.4 Calculating Tbias Using One SV, User Position KnownPerhaps the simplest application of GPS is synchronizing a receiver’s clock when thereceiver’s position is known. In this example we will form an ﬁrst-order estimate ofthe Tbias term associated with receiver clock. Figure 2.4 shows a model of ourexample system. The SV has its own clock and sends its clock timing information tothe earth-based GPS receiver using an encoded radio wave. The receiver has twoclocks, a reference clock and a replica of the SV clock reconstructed from thereceived radio wave. Figure 2.4 shows the receiver clock corrected with the calcu-lated Tbias term, i.e., it is in synchronism with the SV clock.
2.4 Calculating Tbias Using One SV, User Position Known 27 100MSEC 0 EC 20 0M SE MS C 20 C SE 0 1 0.9 77 uSE C GPS SATELLITE 300 MSEC MSEC900 2MSEC 3MSEC 0 4M 0 SE C C = SPEED OF LIGHT EC L) C R = t*C C 5M SE uS SE SE 77 OTA C .9 T M 1M =0 S 20 SEC 400MS C TIC TI 1 023 6M EC 17MSEC 18MS (1 EC EC 7MSEC 800MS 8MSEC t = Trec - Tsent - Tbias SEC 16M C SE 9M C SE M 15 C SE 14 M MS 10 EC SEC C SE 13MS 11M EC 12MSEC C ME S 00 M C 0M SE 50 5 0 70 600MSEC R= (Xu - Xsv)2 + (Yu - Ysv)2 + (Zu - Zsv)2 Tbias = Trec - Tsent - R C SV POSITION (X, Y, Z) t= R Pa th De lay c DKD INSTRUMENTS USER REC. 1 SEC 0 1 SEC 0 20MSEC 20MSEC C 10 C 10 SE 0M SE 0M 0M SE 0M SE 90 C 90 C 1MSEC 0 1MSEC 0 C C 0.977uSEC 0 SE 200M SE 200M 0.977uSEC 0 800M 800M SEC 1 TIC =0.977uSEC SEC (1023 TICS TOTAL) 1 TIC =0.977uSEC (1023 TICS TOTAL) 20MSEC 0 20MSEC 0 C 2M C 2M MSE SE C MSE SE 18 18 C C C MSE 3M MSE 3M SEC SEC 300M SE 300M SE 17 17 C C 700M SEC 700M SEC SEC 4MS SEC 4MS 15MSEC 16M 15MSEC 16M EC EC 5MSEC 5MSEC EC 14M EC 14M 6MS 6MS SEC SEC C C 13 SE 13 SE MSE 7M MSE 7M C 40 40 C C SE C 12 0M MSE C SE 12 C 0M 0M SE SE MSE SE 60 C 8M C 0M SE 11M 60 C 8M C SEC EC 11M EC 9MS SEC 9MS 10MSEC 10MSEC 500MSEC 500MSEC 500MSEC REPLICA OF RECEIVER CLOCK FROM SV REFERENCE CLOCK USER POSITION (Xu, Yu, Zu) EARTHFig. 2.4 Calculation of Tbias using a single SV, user and SV position known In order to solve for Tbias we need to calculate the distance R from the SV to theUser receiver and measure Trec and Tsent as shown in Fig. 2.4. The calculation of Ruses the distance equation which requires the user and SV positions in X, Y, Z. We haveassumed the user position is known. In addition to the SV clock information the SV
28 2 Introduction to the Global Positioning Systemposition information is encoded onto the radio wave that is transmitted to the receiver.We will assume for now that this information is provided in X, Y, Z coordinates.We will see later that getting SV position information in X, Y, Z format is nontrivial. The Tsent and Trec information are obtained as before by just taking a “snapshot” of the receiver reference clock and the receiver’s replica of the SV clock. Ifwe record the SV position at the same instant that we record Trec and Tsent, we willhave all the information needed to solve for Tbias. It is important to realize that dueto SV motion we must “capture” the SV position data at the same moment wecapture the state of the receiver’s clocks. If we do not properly capture SV position,Tsent, and Trec, then the computed distance, R, would be incorrect for the measuredpath delay. Now that we have all the information needed we can calculate R, Dt, and Tbias.If we continually update the measurements and calculations, the receiver referenceclock will “track” the SV clock. This allows the GPS time receiver to replicate thestability of the SV clock. SV clocks are atomic based and so the stability is veryhigh. If we modify the receiver’s reference clock to output a “pulse” every time the1 s dial passes the 0 tic mark and a 1 pps signal will be generated. This is a commonsignal many GPS receivers provide.2.5 GPS Time Receiver Using Master Clock and the Delay term TatmIn the previous example we computed Tbias when the user position was known. Inthis example we will include the effects of SV clock error with respect to the GPSmaster clock and the additional delay caused by diffraction of the radio beam as itpasses through the earth’s atmosphere. Figure 2.5 shows the details of our new model. The receiver’s reference clock isshown corrected to the master clock time. The SV clock has a small error withrespect to master clock. The error is less than a millisecond. As before we capturethe state of the receiver’s reference clock and the replica clock. This information isused in conjunction with the computed path length R to form our estimate of Tbias.There is a difference from our ﬁrst example and that is in the path delay. Theexpression for the path delay now has two additional terms. One, of course, is theSV clock error with respect to the master clock, Terr_sv. The other is the termTatm. The term Tatm is the extra delay experienced by the radio beam as it travelsthrough the earth’s atmosphere. To a ﬁrst approximation the atmosphere acts like alens and “bends” the radio beam from the SV to receiver. This bending of the radiobeam as it passes through the atmosphere causes the extra delay. Normally the delayTatm is broken into two parts, one for the Ionosphere and one for the Tropospherelayers of the earth’s atmosphere. Here we have combined them into one term withthe sign convention following most of all GPS literature. How big is the added