3 Signal Characteristics and Information Extraction Why is the GPS signal so complex? GPS was designed to be readily accessible to millions of military and civilian users. Therefore, it is a receive-only passive system for a user, and the number of users that can simultaneously use the system is unlimited. Because there are many functions that must be performed, the GPS signal has a rather complex structure. As a consequence, there is a correspondingly complex sequence of operations that a GPS receiver must carry out in order to extract desired information from the signal. In this chapter we characterize the signal mathematically, describe the purposes and properties of the important signal components, and discuss generic methods for extracting information from these components. 3.1 MATHEMATICAL SIGNAL WAVEFORM MODELS Each GPS satellite simultaneously transmits on two L-band frequencies denoted by L 1 and L 2 , which are 1575.42 and 1227.60 MHz, respectively. The carrier of the L 1 signal consists of an in-phase and a quadrature-phase component. The in-phase component is biphase modulated by a 50-bps data stream and a pseudorandom code called the C=A-code consisting of a 1023-chip sequence that has a period of 1 ms and a chipping rate of 1.023 MHz. The quadrature-phase component is also biphase modulated by the same 50-bps data stream but with a different pseudorandom code 30 Global Positioning Systems, Inertial Navigation, and Integration, Mohinder S. Grewal, Lawrence R. Weill, Angus P. Andrews Copyright # 2001 John Wiley & Sons, Inc. Print ISBN 0-471-35032-X Electronic ISBN 0-471-20071-9 called the P-code, which has a 10.23-MHz chipping rate and a one-week period. The mathematical model of the L 1 waveform is st  2P I p dtct cosot  y  2P Q q dtpt sinot  y; 3:1 where P I and P Q are the respective carrier powers for the in-phase and quadrature- phase carrier components, dt is the 50-bps data modulation, ct and pt are the respective C=A and P pseudorandom code waveforms, o is the L 1 carrier frequency in radians per second, and y is a common phase shift in radians. The quadrature carrier power P Q is approximately 3 dB less than P I . In contrast to the L 1 signal, the L 2 signal is modulated with only the 50-bps data and the P-code, although there is the option of not transmitting the 50-bps data stream. The mathematical model of the L 2 waveform is st  2P Q q dtpt sinot  y: 3:2 Figures 3.1 and 3.2 respectively show the structure of the in-phase and quadrature- phase components of the L 1 signal. The 50-bps data bit boundaries always occur at 1 bit (20 ms) 1ms 1 chip (0.9775 µs) d(t) 50 bps data 2P I cos(ω t) L 1 Carrier c(t) C/A code Multiply Multiply Transmitted signal Data 50 bps 1bit=20ms C/A-code epochs 20 code periods per bit Oneperiod(1ms) of C/A code 1023 chips/ period L 1 Carrier 1575.42 MHz 1540 cycles/chip 1 0 2 3 4 5 6 7 17 18 19 20 Fig. 3.1 Structure of in-phase component of the L 1 signal. 3.1 MATHEMATICAL SIGNAL WAVEFORM MODELS 31 an epoch of the C=A-code. The C=A-code epochs mark the beginning of each period of the C=A-code, and there are precisely 20 code epochs per data bit, or 20,460 C=A-code chips. Within each C=A-code chip there are precisely 1540 L 1 carrier cycles. In the quadrature-phase component of the L 1 signal there are precisely 204,600 P-code chips within each 50-bps data bit, and the data bit boundaries always coincide with the beginning of a P-code chip [42, 56]. 3.2 GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES 3.2.1 50-bps Data Stream The 50-bps data stream conveys the navigation message, which includes, but is not limited to, the following information: 1. Satellite Almanac Data. Each satellite transmits orbital data called the almanac, which enables the user to calculate the approximate location of every satellite in the GPS constellation at any given time. Almanac data is not accurate enough for determining position but can be stored in a receiver where it remains valid for many months. It is primarily used to determine which satellites are visible at a given location so that the receiver can search for those satellites when it is ®rst turned on. It can also be used to determine the 1 bit (20 ms) 1 chip (0.09775 µs) d(t) 50 bps data 2P Q sin(ωt) L 1 Carrier ρ(t) P(Y )-code Multiply Multiply Transmitted signal Data 50 bps 1bit=20ms 204,600 chips (20 ms) of P(Y )-code Period = 1 week (≅ 6.19 × 10 12 chips) L 1 carrier 1575.42 MHz 154 cycles/chip Fig. 3.2 Structure of quadrature-phase component of the L 1 signal. 32 SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION approximate expected signal Doppler shift to aid in rapid acquisition of the satellite signals. 2. Satellite Ephemeris Data. Ephemeris data is similar to almanac data but enables a much more accurate determination of satellite position needed to convert signal propagation delay into an estimate of user position. In contrast to almanac data, ephemeris data for a particular satellite is only broadcast by that satellite, and the data is valid for only several hours. 3. Signal Timing Data. The 50-bps data stream includes time tagging, which is used to establish the transmission time of speci®c points on the GPS signal. This information is needed to determine the satellite-to-user propagation delay used for ranging. 4. Ionospheric Delay Data. Ranging errors due to ionospheric effects can be partially canceled by using estimates of ionospheric delay that are broadcast in the data stream. 5. Satellite Health Message. The data stream also contains information regarding the current health of the satellite, so that the receiver can ignore that satellite if it is not operating properly. Structure of the Navigation Message The information in the navigation message has the basic frame structure shown in Fig. 3.3. A complete message consists of 25 frames, each containing 1500 bits. Each frame is subdivided into ®ve 300-bit subframes, and each subframe consists of 10 words of 30 bits each, with the most signi®cant bit (MSB) of the word transmitted ®rst. Thus, at the 50-bps rate it takes 6 s to transmit a subframe and 30 s to complete one frame. Transmission of the complete 25-frame navigation message requires 750 s, or 12.5 min. Except for occasional updating, subframes 1, 2, and 3 are constant (i.e., repeat) with each frame at the 30-s frame repetition rate. On the other hand, subframes 4 and 5 are each subcommutated 25 times. The 25 versions of subframes 4 and 5 are referred to as pages 1±25. Hence, except for occasional updating, each of these pages repeats every 750 s, or 12.5 min. A detailed description of all information contained in the navigation message is beyond the scope of this text. Therefore, we only give an overview of the fundamental elements. Each subframe begins with a telemetry word (TLM). The ®rst 8 bits of the TLM is a preamble that makes it possible for the receiver to determine when a subframe begins. The remainder of the TLM contains parity bits and a telemetry message that is available only to authorized users and is not a fundamental item. The second word of each subframe is called the hand-over word (HOW). Z-Count Information contained in the HOW is derived from a 29-bit quantity called the Z-count. The Z-count is not transmitted as a single word, but part of it is transmitted within the HOW. The Z-count counts epochs generated by the X 1 register of the P-code generator in the satellite, which occur every 1.5 s. The 19 LSBs of the Z-count, called the time-of-week (TOW) count, indicate the number of X 1 epochs 3.2 GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES 33 that have occurred since the start of the current week. The start of the current week occurs at the X 1 epoch, which occurs at approximately midnight of Saturday night=Sunday morning. The TOW count increases from zero at the start of the week to 403,199 and then rolls over to zero again at the start of the following week. A TOW count of zero always occurs at the beginning of subframe 1 of the ®rst frame (the frame containing page 1 of subcommutated subframes 4 and 5). A truncated version of the TOW count, containing its 17 MSBs, comprises the ®rst 17 bits of the HOW. Multiplication of this truncated count by 4 gives the TOW count at the start of the following subframe. Since the receiver can use the TLM preamble to determine precisely the time at which each subframe begins, a method for determining the time of transmission of any part of the GPS signal is thereby established. The relationship between the HOW counts and TOW counts is shown in Fig. 3.4. GPS Week Number The 10 MSBs of the Z-count contain the GPS week number (WN), which is a modulo-1024 week count. The zero state is de®ned to be that week that started with the X 1 epoch occurring at approximately midnight on the night of January 5, 1980=morning of January 6, 1980. Because WN is a modulo-1024 count, an event called the week rollover occurs every 1024 weeks (a few months short of 20 years), and GPS receivers must be designed to accommodate it. 1 The WN is not part of the HOW but instead appears as the ®rst 10 bits of the third word in subframe 1. Fig. 3.3 Navigation message frame structure. 1 The most recent rollover occurred at GPS time zero on August 22, 1999, with little dif®culty. 34 SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION Frame and Subframe Identi®cation Three bits of the HOW are used to identify which of the ®ve subframes is being transmitted. The frame being transmitted (corresponding to a page number from 1 to 25) can readily be identi®ed from the TOW count computed from the HOW of subframe 5. This TOW count is the TOW at the start of the next frame. Since there are 20 TOW counts per frame, the frame number of that frame is simply (TOW=20) (mod 25). Information by Subframe In addition to the TLM and HOW, which occur in every subframe, the following information is contained within the remaining eight words of subframes 1±5 (only fundamental information is described): 1. Subframe 1. The WN portion of the Z-count is part of word 3 in this subframe. Subframe 1 also contains GPS clock correction data for the satellite in the form of polynomial coef®cients de®ning how the correction varies with time. Time de®ned by the clocks in the satellite is commonly called SV time (space vehicle time); the time after corrections have been applied is called GPS time. Thus, even though individual satellites may not have perfectly synchronized SV times, they do share a common GPS time. Additional information in subframe 1 includes the quantities t 0c , T GD , and IODC. The clock reference time t 0c is used as a time origin to calculate satellite clock error, the ionospheric group delay T GD is used to correct for ionospheric propagation delay errors, and IODC (issue of date, clock) indicates the issue number of the clock data set to alert users to changes in clock parameters. 2. Subframes 2 and 3. These subframes contain the ephemeris data, which is used to determine the precise satellite position and velocity required by the Fig. 3.4 Relationship between HOWcounts and TOWcounts. 3.2 GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES 35 navigation solution. Unlike the almanac data, this data is very precise, is valid over a relatively short period of time (several hours), and applies only to the satellite transmitting it. The components of the ephemeris data are listed in Table 3.1, and the algorithm that should be used to compute satellite position in WGS 84 coordinates is given in Table 3.2. The satellite position computa- tion using these data is implemented in the Matlab m-®le ephemeris.m on the accompanying diskette. The IODE (issue of date, ephemeris) informs users when changes in ephemeris parameters have occurred. Each time new parameters are uploaded from the GPS control segment, the IODE number changes. 3. Subframe 4. The 25 pages of this subframe contain the almanac for satellites with PRN (pseudorandom code) numbers 25 and higher, as well as special messages, ionospheric correction terms, and coef®cients to convert GPS time to UTC time. There are also spare words for possible future applications. The components of an almanac are very similar to those of the ephemeris, and the calculation of satellite position is performed in essentially the same way. Table 3.1 Components of Ephemeris Data Name Description Units a M 0 Mean anomaly at reference time semicircle Dn Mean motion difference from computed value semicircle=s e Eccentricity dimensionless  a p Square root of semimajor axis m 1=2 O 0 Longitude of ascending node of orbit plane at weekly epoch semicircle i 0 Inclination angle at reference time semicircle o Argument of perigee semicircle _ O Rate of right ascension semicircle=s IDOT Rate of inclination angle semicircle=s C uc Amplitude of cosine harmonic correction term to the rad argument of latitude C us Amplitude of sine harmonic correction term to the rad argument of latitude C rc Amplitude of cosine harmonic correction term to the m orbit radius C rs Amplitude of sine harmonic correction term to the m orbit radius C ic Amplitude of cosine harmonic correction term to the rad angle of inclination C is Amplitude of sine harmonic correction term to the rad angle of inclination t 0e Ephemeris reference time s IODE Issue of data, ephemeris dimensionless a Units used in MATLAB m-®le ephemeris are different. 36 SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION 4. Subframe 5. The 25 pages of this subframe includes the almanac for satellites with PRN numbers from 1 to 24. It should be noted that since each satellite transmits all 25 pages, almanac data for all satellites is transmitted by every satellite. Unlike ephemeris data, almanac data is valid for long periods (months) but is much less precise. Additional data contained in the navigation message is user range error (URE), which estimates the range error due to errors in satellite ephemeris, timing errors, and selective availability (SA) and ¯ags to indicate the health status of the satellites. Table 3.2 Algorithm for Computing Satellite Position m  3:986005  10 14 m 3 =s 2 WGS 84 value of earth's universal gravitational parameter _ O e  7:292115167 10 À5 rad=s WGS 84 value of earth's rotation rate a   a p  2 Semimajor axis n 0   m=a 3 p Computed mean motion, rad=s t k  t À t 0e a Time from ephemeris reference epoch n  n 0  D n Corrected mean motion M k  M 0  nt k Mean anomaly M k  E k À e sin E k Kepler's equation for eccentric anomaly f k  cos À1 cos E k À 1 1 À e cos E k  True anomaly from cosine f k  sin À1  1 À e 2 p sin E k 1 À e cos E k ! True anomaly from sine E k  cos À1 e  cos f k 1  e cos f k  Eccentric anomaly from cosine f k  f k  o Argument of latitude dm k  C mc cos 2f k  C ms sin 2f k Second-harmonic correction to argument of latitude dr k  C rc cos 2f k  C rs sin 2f k Second-harmonic correction to radius di k  C ic cos 2f k  C is sin 2f k Second-harmonic correction to inclination m k  f k  dm k Corrected argument of latitude r k  a1 À e cos E k dr k Corrected radius i k  i 0  di k IDOTt k Corrected inclination x H k  r k cos m k X coordinate in orbit plane y H k  r k sin m k Y coordinate in orbit plane O k  O 0  _ O À _ O 0 t k À _ O e t 0e Corrected longitude of ascending node x k  x H k cos O k À y H k cos i k sin O k ECEF X coordinate y k  x H k sin O k  y H k cos i k cos O k ECEF Y coordinate z k  y H k sin i k ECEF Z coordinate a t is in GPS system time at time of transmission, i.e., GPS time corrected for transit time (range=speed of light). Furthermore, t k shall be the actual total time difference between the time t and the time epoch t 0e and must account for beginning or end of week crossovers. That is, if t k is greater than 302,400 s, subtract 604,800 s from t k .Ift k is less than À302; 400 s, add 604,800 s to t k . 3.2 GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES 37 3.2.2 C=A-Code and Its Properties The C=A-code has the following functions: 1. To enable accurate range measurements and resistance to errors caused by multipath. To establish the position of a user to within 10±100 m, accurate user-to-satellite range estimates are needed. The estimates are made from measurements of signal propagation delay from the satellite to the user. To achieve the required accuracy in measuring signal delay, the GPS carrier must be modulated by a waveform having a relatively large bandwidth. The needed bandwidth is provided by the C=A-code modulation, which also permits the receiver to use correlation processing to effectively combat measurement errors due to thermal noise. Because the C=A-code causes the bandwidth of the signal to be much greater than that needed to convey the 50-bps data stream, the resulting signal is called a spread-spectrum signal. Using the C=A-code to increase the signal bandwidth also reduces errors in measuring signal delay caused by multipath (the arrival of the signal via multiple paths such as re¯ections from objects near the receiver antenna) since the ability to separate the direct path signal from the re¯ected signal improves as the signal bandwidth is made larger. 2. To permit simultaneous range measurement from several satellites. The use of a distinct C=A-code for each satellite permits all satellites to use the same L 1 and L 2 frequencies without interfering with each other. This is possible because the signal from an individual satellite can be isolated by correlating it with a replica of its C=A-code in the receiver. This causes the C=A-code modulation from that satellite to be removed so that the signal contains only the 50-bps data and is therefore narrow band. This process is called despreading of the signal. However, the correlation process does not cause the signals from other satellites to become narrow band, because the codes from different satellites are orthogonal. Therefore the interfering signals can be rejected by passing the desired despread signal through a narrow-band ®lter, a bandwith-sharing process called code division multiplexing (CDM) or code division multiple access (CDMA). 3. To provide protection from jamming. The C=A-code also provides a measure of protection from intentional or unintentional jamming of the received signal by another man-made signal. The correlation process that despreads the desired signal has the property of spreading any other signal. Therefore, the signal power of any interfering signal, even if it is narrow band, will be spread over a large frequency band, and only that portion of the power lying in the narrow-band ®lter will compete with the desired signal. The C=A-code provides about 20±30 dB of improvement in resistance to jamming from narrowband signals. We next detail important properties of the C=A-code. 38 SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION Temporal Structure Each satellite has a unique C=A-code, but all of the codes consist of a repeating sequence of 1023 chips occurring at a rate of 1.023 MHz with a period of 1 ms, as previously shown in Fig. 3.1. The leading edge of a speci®c chip in the sequence, called the C=A-code epoch, de®nes the beginning of a new period. Each chip is either positive or negative with the same magnitude. The polarities of the 1023 chips appear to be randomly distributed but are in fact generated by a deterministic algorithm implemented by shift registers. The algorithm produces maximal-length Gold codes, which have the property of low cross-correlation between different codes (orthogonality) as well as reasonably small autocorrelation sidelobes. Autocorrelation Function The autocovariance 2 function of the C=A-code is ct 1 T  T 0 ctct À t dt; 3:3 where ct is the idealized C=A-code waveform (with chip values of Æ1), t is the relative delay measured in seconds, and T is the code period (1 ms). The auto- correlation function is periodic in t with a period of 1 ms. A single period is plotted in Fig. 3.5, which is basically a triangle two chips wide at its base with a peak located at t  0 [in reality ct contains small-sidelobe structures outside the triangular region, but these are of little consequence]. The C=A-code autocorrelation function plays a substantial role in GPS receivers, inasmuch as it forms the basis for code tracking and accurate user-to-satellite range Fig. 3.5 Autocorrelation functions of C=A- and P(Y)-codes. 2 Strictly speaking, the autocorrelation function ctct=c0 is the autocovariance function rescaled by the signal variance [c0], but the terms autocorrelation and autocovariance are often interchanged in engineering usage. 3.2 GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES 39 [...]... The Y-code is formed by multiplying the P-code by an encrypting code called the W-code The W-code is a random-looking sequence of chips that occur at a 511.5-kHz rate Thus there are 20 P-code chips for every W-code chip Since both the P-code and the W-code have chip values of Æ1, the resulting Y-code has the same appearance as the P-code, that is, it also has a 10.23-MHz chipping rate However, the Y-code... the P-code is so long, the power spectrum may be regarded as continuous for practical purposes Each satellite broadcasts a unique P-code The technique used to generate it is similar to that of the C=A-code, but somewhat more complicated, and will not be covered in this book Y-Code The encrypted form of the P-code used for antispoo®ng and denial of the P-code to unauthorized users is called the Y-code... C=A rate, and it has a period of one week It is transmitted synchronously with the C=A- code in the sense that each chip transition of the C=A-code always corresponds to a chip transition in the P-code Like the C=A-code, the P-code autocorrelation function has a triangular central peak centered at t ˆ 0, but with one-tenth the base width, as shown in Fig 3.5 The power spectrum also has a sin2 …x†=x2... appears to be a GPS signal (spoo®ng), but in reality is designed to confuse the GPS receiver This is prevented by encrypting the P-code The would-be spoofer cannot know the encryption process and cannot make the contending signal look like a properly encrypted signal Thus the receiver can reject the false signal and decrypt the desired one 3 Denial of P-Code Use The structure of the P-code is published... improves as the signal bandwidth increases Thus, the P-code provides improved range measurement accuracy as compared to the C=A-code Simultaneous range measurements using both codes is even better Due to its increased bandwidth, the P-code is also more resistant to range errors caused by multipath P-Code Characteristics Unlike the C=A-code, the P-code modulates both the L1 and L2 carriers Its chipping... fact that the values of the ideal C=A-code waveform are Æ1 (in reality the received C=A-code waveform is not ideal due to bandlimiting in the receiver; however, the effects are usually minor) This procedure, called code despreading, removes the C=A-code modulation from the signal The resulting signal has a two-sided spectral width of approximately 100 Hz due to the 50-bps data modulation From the above... for comparative purposes is a typical noise power spectral density found in a GPS receiver after – – – – Fig 3.6 Power spectra of C=A- and P(Y)-codes 3.2 GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES 41 frequency conversion of the signal to baseband (i.e., with carrier removed) It can be seen that the presence of the C=A-code causes the entire signal to lie well below the noise level, because the... is, it also has a 10.23-MHz chipping rate However, the Y-code cannot be despread by a receiver replica P-code unless it is decrypted Decryption consists of multiplying the Y-code by a receiver-generated replica of the W-code which is made available only to authorized users Since the encrypting W-code is also not known by the creators of spoo®ng signals, it is easy to verify that such signals are not... Search for Signal in Frequency and C=A-Code Phase Why is a Signal Search Necessary? Since GPS signals are radio signals, one might assume that they could be received simply by setting a dial to a particular frequency, as is done with AM and FM broadcast band receivers Unfortunately, this is not the case 1 GPS signals are spread-spectrum signals in which the C=A or P-codes spread the total signal power... following functions: 1 Increased Jamming Protection Because the bandwidth of the P-code is 10 times greater than that of the C=A-code, it offers approximately 10 dB more protection from narrow-band interference In military applications the interference is likely to be a deliberate attempt to jam (render useless) the received GPS signal 2 Provision for Antispoo®ng In addition to jamming, another military . Sons, Inc. Print ISBN 0-4 7 1-3 5032-X Electronic ISBN 0-4 7 1-2 007 1-9 called the P-code, which has a 10.23-MHz chipping rate and a one-week period. The mathematical. the P-code to unauthorized users is called the Y-code. The Y-code is formed by multiplying the P-code by an encrypting code called the W-code. The W-code