Gps

Chapter 1
GLOBAL POSITIONING SYSTEM
He-Sheng Wang
Department of Communications and Guidance Engineering
National Taiwan Ocean University
Taipei, Taiwan
Abstract The Global Positioning System (GPS) is a space-based radionavigation
system managed and operated by the United States (U.S.) Government.
Through the developments of a few decades, GPS is now a very well-
known public science. The objective of this chapter is to provide a
compact, technical overview of GPS. Many aspects about GPS will be
discussed including the system, signals, measurements, performance,
and applications.
Keywords: Global Positioning System, Standard Positioning Service, Precise Posi-
tioning Service, Pseudo Random Noise, Pseudorange, GPS Observables,
Code Division Multiple Access, Spread Spectrum, Binary Phase Shift
Keying
1. What is Navigation?
Navigation is without a doubt one of the oldest scientific problems
in the world. Since prehistoric times, people have been trying to figure
out a satisfactory way to tell where they are and to help guide them
to where they are going. Cavemen probably used stones and twigs to
mark a trail when they set out hunting for food. The earliest mariners
followed the coast closely to prevent from getting lost. When navigators
first sailed into the open ocean, they discovered they could chart their
course by following the stars. The ancient Phoenicians used the North
Star to journey from Egypt and Crete. According to Homer, the god-
dess Athena told Odysseus to keep the Great Bear on his left during his
travels from Calypsos Island. Unfortunately, the stars are only visible
2 Fundamentals of Electrical Engineering
at night - and only on clear nights.
The next major developments in the quest for the perfect method of
navigation were the magnetic compass and the sextant. The needle of
a compass always points north, so it is always possible to know in what
direction you are going. The sextant uses adjustable mirrors to measure
the exact angle of the stars, moon, and sun above the horizon.
In the early 20th century several radio-based navigation systems were
developed, which were used widely during World War II. Both allied and
enemy ships and airplanes used ground-based radio-navigation systems
as the technology advanced. One drawback of using radio waves gener-
ated on the ground is that you must choose between a system that is
very accurate but doesnt cover a wide area, or one that covers a wide
area but is not very accurate. High-frequency radio waves (like UHF
TV) can provide accurate position location but can only be picked up
in a small, localized area. Lower frequency radio waves (like AM radio)
can cover a larger area, but are not a good yardstick to tell you exactly
where you are.
Scientists, therefore, decided that the only way to provide coverage
for the entire world was to place high-frequency radio transmitters in
space. A transmitter high above the Earth sending a high-frequency
radio wave with a special coded signal can cover a large area and still
overcome much of the noise encountered on the way to the ground. This
is one of the main principles behind the GPS system.
GPS is actually a property of the U.S. government. It was developed
under the supervision of the U.S. Department of Defense (DoD). GPS
was designed as a dual-use system with the primary purpose of enhanc-
ing the effectiveness of U.S. and allied military forces. It is expected
to provide a substantial military advantage and is now being integrated
into virtually every facet of military operations. GPS is also becoming an
integral component of the Global Information Infrastructure, with appli-
cations ranging from mapping and surveying to international air traffic
management and global climate change research. In an effort to make
this beneficial service available to the greatest number of users while
ensuring that the national security interests of the United States are ob-
served, two GPS services are provided. The Precise Positioning Service
(PPS) is available primarily to the military and other authorized users
of the United States and its allies equipped with PPS receivers. The
Standard Positioning Service (SPS) was originally designed to provide
Global Positioning System 3
civil users with a less accurate positioning capability than PPS through
the use of a technique known as Selective Availability (SA). On May 1,
2000, the President directed the U.S. Department of Defense (DoD) to
discontinue the use of SA effective midnight May 1, 2000.
2. The Basic GPS Concept
The method of radionavigation is parcitularly attractive mainly be-
cause its relatively simple principle. The radio waves travel at a know
speed. Therefore, if the transit time of a signal from a transmitting
station can be measured, the distance between the transmitter and the
observer can be determined. Given distances to three transmitters at
known locations, the observer can compute his position unambiguously.
Estimation of a position based on measurement of distances is referred
to as trilateration. A radionavigation system based on this idea is re-
ferred to as a time-of-arrival (TOA) system. GPS is a TOA system.
Figure 1.1 shows a two-dimensional case. In order to determine the
user position, three satellites and three distances are required. The trace
of a point with constant distance to a fixed point is a circle in the two-
dimensional case. Two satellites and two distances give two possible
solutions because two circles intersect at two points. A third circle is
needed to uniquely determine the user position.
For similar reasons one might decide that in a three-dimensional case
four satellites and four distances are needed. The equal-distance trace
to a fixed point is a sphere in a three-dimensional case. Two spheres in-
tersect to make a circle. This circle intersects another sphere to produce
two points. In order to determine which point is the user position, one
more satellite is needed.
In GPS the position of the satellite is known from the ephemeris data
transmitted by the satellite. One can measure the distance from the
receiver to the satellite. Therefore, the position of the receiver can be
determined. In the above discussion, the distance measured from the
user to the satellite is assumed to be very accurate and there is no bias
error. However, the distance measured between the receiver and the
satellite has a constant unknown bias, because the user clock usually
is different from the GPS clock. In order to resolve this bias error one
more satellite is required. Therefore, in order to find the user position
4 Fundamentals of Electrical Engineering
Figure 1.1. Two-dimensional user position
five satellites are needed.
If one uses four satellites and the measured distance with bias error
to measure a user position, two possible solutions can be obtained. The-
oretically, one cannot determine the user position. However, one of the
solutions is close to the earths surface and the other one is in space.
Since the user position is usually close to the surface of the earth, it
can be uniquely determined. Therefore, the general statement is that
four satellites can be used to determine a user position, even though the
distance measured has a bias error.
3. System Architecture[12]
The GPS system is comprised of three segments, whose purpose is to
provide a reliable and continuous positioning and timing service to the
GPS user community. These three segments are known as the Space
Segment, Control Segment, and User Segment. See Figure 1.2. The
Global Positioning System 5
Space Segment consists of 24 satellites, each in its own orbit 26,560
km above the Earth. The Control Segment consists of ground stations
(five of them, located around the world) that make sure the satellites
are working properly. The DoD is responsible for both the Space and
Control Segments. The User Segment is comprised of receivers from a
wide variety of international agencies (a relative large part of them is
from U.S., of course), in addition to the growing private user base[2].
Figure 1.2. GPS consists of three major segments: the Space Segment, the Control
Segment, and the User Segment.
3.1 The GPS Space Segment
The GPS space segment consists nominally of a constellation of 24
operational Block II satellites (Block II, IIA, and IIR). The GPS satel-
lite constellation is shown in Figure 1.3 The satellites are arranged in six
orbital planes inclined at 55 degree relative to the equatorial plane, with
four primary satellite slots distributed unevenly in each orbit. Charac-
teristics of GPS satellites are summarized in Table 1.1. Each satellite
broadcasts a navigation message based upon data periodically uploaded
from the Control Segment and adds the message to a 1.023 MHz Pseudo
Random Noise (PRN) Coarse/Acquisition (C/A) code sequence. The
6 Fundamentals of Electrical Engineering
Table 1.1. Characteristics of GPS Satellites
Constellation
Number of Satellites 24
Number of Orbital Planes 6
Number of Satellites Per Orbit 4
Orbital Inclination 55
o
Orbital Radius 26560 km
Period 11 hrs 57 min 57.26 sec
Ground Track Repeat Sideral Day
satellite modulates the resulting code sequence onto a 1575.42 MHz L-
band carrier to create a spread spectrum ranging signal, which it then
broadcasts to the user community. This broadcast is known as the SPS
ranging signal or the L1 signal. Each C/A code is unique, and pro-
vides the mechanism to identify each satellite in the constellation. A
block diagram illustrating the Block IIA satellite’s SPS ranging signal
generation process is provided in Figure 1.4. The GPS satellite also
transmits a second ranging signal known as L2, that supports PPS user
two-frequency corrections. L2, like L1, is a spread spectrum signal and
is transmitted at 1227.6 MHz.
The Block II satellites are designed to provide reliable service over a
7.5- to 10-year design life, depending on the production version, through
a combination of space qualified parts, multiple redundancies for critical
subsystems, and internal diagnostic logic. The Block II satellite requires
minimal interaction with the ground and allows all but a few mainte-
nance activities to be conducted without interruption to the ranging
signal broadcast. Periodic uploads of data to support navigation mes-
sage generation are designed to cause no disruption to the SPS ranging
signal, although Block II/IIA satellites may experience a 6- to 24-second
disruption upon transition to the new upload.
3.2 The GPS Control Segment
The GPS Control Segment (CS) is comprised of four major compo-
nents: a Master Control Station (MCS), Backup Master Control Station
(BMCS), four ground antennas, and six monitor stations. An overview
of the CS is provided in Figure 1.5. The MCS is located at Schriever Air
Force Base, Colorado, and is the central control node for the GPS satel-
lite constellation. Operations are maintained 24 hours a day, seven days
Global Positioning System 7
Figure 1.3. The GPS Space Segment consists of a baseline constellation of 24 satel-
lites distributed in six orbital planes.
a week throughout each year. The MCS is responsible for all aspects of
constellation command and control, to include:
Routine satellite bus and payload status monitoring.
Satellite maintenance and anomaly resolution.
Managing SPS performance in support of all performance stan-
dards.
Navigation data upload operations as required to sustain perfor-
mance in accordance with accuracy performance standards.
Prompt detection and response to service failures.
In the event of a prolonged MCS outage, GPS operations can be
moved to a contractor-owned BMCS located at Gaithersburg, MD. When
required, personnel from the MCS deploy to the BMCS within 24 hours.
The BMCS is operationally exercised approximately four times per year
to ensure system capability. The CS’s four ground antennas provide a
8 Fundamentals of Electrical Engineering
Figure 1.4. Block IIA SPS Ranging Signal Generation and Transmission
near real-time Telemetry, Tracking, and Commanding (TT&C) inter-
face between the GPS satellites and the MCS. The six monitor stations
provide near real-time satellite ranging measurement data to the MCS
and support near-continuous monitoring of constellation performance.
The current CS monitor stations provide approximately 93% global cov-
erage, with all monitor stations operational, with a 5 degree elevation
mask angle. The actual elevation angle that a monitor station acquires
any given satellite varies due to several external factors.
3.3 The GPS User Segment
The GPS User Segment consists of the GPS receivers and the user
community. GPS receivers convert satellite signals into position, veloc-
ity, and time estimates. Four satellites are required to compute the four
dimensions of X, Y, Z (position) and Time. GPS receivers are used for
navigation, positioning, time dissemination, and other research. Because
new receivers are continuously being developed, it seems inappropriate to
Global Positioning System 9
Figure 1.5. The GPS Control Segment
expend much effort in describing individual receivers in detail. However,
the technology inside the receiver is intellectually challenging, stimulat-
ing, and truely in line with today’s hige tech.
A partial listing of the applications of GPS including the following:
Aircraft navigation - GPS and differential GPS, commercial and
general aviation aircraft
Land mobile navigation - automobiles, trucks, and buses
Marine vessel navigation - GPS and differential GPS
Time transfer between clocks
Spacecraft orbit determination
Attitude determination using multiple antennas
Kinematic survey
Ionospheric measurement
10 Fundamentals of Electrical Engineering
4. Signals
Currently, the GPS satellite transmits two Right Hand Circularly Po-
larized (RHCP) L-band signals known as Link 1 (L1) and Link 2 (L2).
The L-band covers frequencies between 1 GHz and 2 GHz, and is a sub-
set of the ultra-high frequency (UHF) band. The center frequencies of
L1 and L2 are as follows:
L1 : f
L1
= 1575.42MHz, L2 : f
L2
= 1227.60MHz.
Two signals are transmitted on L1, one for civil users, and the other for
U.S. government-authorized users. L2 transmits only one signals, which
is intended for the authorized users only. Each signal consists of three
components:
Carrier: RF sinusoidal signal with frequency f
L1
or f
L2
.
Ranging Code: a unique sequence of 0s and 1s assigned to each
satellite which allows the receiver to determine the signal transit
time instantaneously.
Navigatioln Message: a binary-coded message consisting of data
on the satellite health status, ephemeris (satellite position and
velocity), clock bias parameters, and an almanac given reduced-
precision ephemeris data on all satellites in the constellation.
As the name implies, the carrier waves provide the means by which
the ranging codes and Navigation Message is transmitted to earth (and
hence to the user). The primary function of the ranging codes is to per-
mit the signal transit time (from satellite to receiver) to be determined.
(This quantity is also sometimes referred to in the navigation literature
as the time-of-arrival – TOA.) The transit time when multiplied by the
speed of electromagnetic radiation (c = 299792458 m/s in a vacuum)
gives the receiver-satellite range. The Navigation Message is modulated
on both carrier frequencies and contains the satellite ephemeris, satel-
lite clock parameters, and other pertinent information such as general
system status messages and an ionospheric delay model, necessary for
real-time navigation to be performed.
Global Positioning System 11
4.1 Code-Division Multiple Access (CDMA)
Signals
In the communication theory, a signal S can be generally written in
the following form:
S = Asin(2πft + φ) (1.1)
where A is the amplitude, f is the frequency, and φ is the initial
phase. These three parameters can be modulated to carry information.
If A is modulated, it is referred to as amplitude modulation. If f is
modulated, it is frequency modulation. If φ is modulated, it is phase
modulation. The GPS signal is a phase-modulated signa with φ = 0, π;
this type of phase modulation is referred to as binary phase shift keying
(BPSK). The phase change rate is often referred to as the chip rate. The
spectrum shape can be described by the sinc function (sinx/x) with the
spectrum width proportional to the chip rate. For example, the chip
rate of C/A-code is 1.023 MHz, therefore the main lobe of the spectrum
has a null-to-null width over 2 MHz. The chip rate of P-code is 10.23
MHz so that its band width is about 20.46 MHz. This type of signal is
also referred to as a spread-spectrum signal. If the modulation code is a
digital sequence with a frequency higher than the data rate, the system
can be called a direct-sequence modulated system. Spectra of the GPS
C/A-code and P-code are shown in Figure 1.6.
Figure 1.6. Power spectra of signals transmitted by a GPS satellite. The energy of
the signal for civil users carrying a C/A-code on L1 is spread mainly over a 2-MHz-
wide frequency band. The bandwidths of the signals for military users on L1 and L2
carrying a P(Y)-code are ten times wider.
12 Fundamentals of Electrical Engineering
A code division multiple access (CDMA) signal in general is a spread-
spectrum system. All the signals in the system use the same center
frequency. The signals are modulated by a set of orthogonal (or near-
orthogonal) codes. In order to acquire an individual signal, the code of
that signal must be used to correlate with the received signal. The GPS
signal is CDMA using direct sequence to bi-phase modulate the carrier
frequency. Since the CDMA signals all use the same carrier frequency,
there is a possibility that the signals will interfere with one another. This
effect will be more prominent when strong and weak signals are mixed
together. In order to avoid the interference, all the signals should have
approximately the same power levels at the receiver. Sometimes in the
acquisition one finds that a cross-correlation peak of a strong signal is
stronger than the desired peak of a weak signal. Under this condition,
the receiver may obtain wrong information.
4.2 Ranging Codes
In a GPS satellite, all signal components are derived from the output
of a highly stable atomic clock. In the operational GPS system each
satellite is equipped with two cesium and two rubidium atomic clocks.
(The Block IIF satellites may be equipped with a space-qualified hy-
drogen maser.) The clocks generate a pure sine wave at a frequency
f
0
= 10.23 MHz, with a stability of the order of 1 part in 1013 over one
day. This is referred to as the fundamental frequency. Multiplying the
fundamental frequency f
0
by integer factors yields the two microwave
L-band carrier waves L1 and L2 respectively. The frequency of the two
waves are obtained as follows:
f
L1
= f
0
×154 = 1575.42MHz
equivalent wavelength: L1 = c/f
L1
≈ 19cm
f
L2
= f
0
×120 = 1227.60MHz
equivalent wavelength: L2 = c/f
L2
≈ 24cm
These are righthand circularly polarized radio frequency waves capa-
ble of transmission through the atmosphere over great distances, but
they contain no information. All satellites broadcast the same frequen-
cies (though the received frequencies are slightly different because of the
Doppler shift). In order to give the carriers information they must be
modified, or modulated, in some way. In the Global Positioning Sys-
tem there are two distinct codes used to modulate the L-band carriers,
namely the ranging codes and the Navigation Message. As described in
Global Positioning System 13
Table 1.2. Power Level of GPS Signals
P C/A
L1 -133 dBm -130 dBm
L2 -136 dBm -136 dBm
∗
*Currently not in L2 frequency
the previous section, the specific form of modulation used is BPSK: a 0
bit leaves the carrier signal unchanged; and a 1 bit multiplies the carrier
by -1, which is equivalent to shifting the phase of the sinusoidal signal
by 180
o
. At code transition from 0 to 1, or from 1 to 0, the phase of the
carrier is shifted by 180
o
.
Two ranging codes used in the satellites are given as follows:
The C/A code, the ”clear/access” or ”coarse/acquisition” code
(sometimes also referred to as the ”S code”).
The P code, the ”private” or ”precise” code, which under Anti-
Spoofing (AS) is replaced by the ”Y” code.
From the above discussion, the signal leaving the antenna of the k-th
satellite can be modeled as:
s
(k)
=
√
2P
C
x
(k)
(t)D
(k)
(t) sin(2πf
L1
t + θ
L1
)
+
_
2P
Y,L1
y
(k)
(t)D
(k)
(t) cos(2πf
L1
t + θ
L1
)
+
_
2P
Y,L2
y
(k)
(t)D
(k)
(t) cos(2πf
L2
t + θ
L2
)
(1.2)
where P
C
is the signal power of C/A-code, P
Y,L1
, and P
Y,L2
are the
signal powers of P(Y)-code on L1 and L2, respectively; x
(k)
(t) = ±1
and y
(k)
(t) = ±1 represent the C/A-code and P(Y)-code sequences,
repsectively, assigned to satellite number k; D
(k)
(t) = ±1 denotes the
navigation data bit stream; f
L1
and f
L2
are the carrier frequencies cor-
reponding to L1 and L2, respectively; θ
L1
and θ
L2
are the initial phase
offsets.
The minimum power levels of the signals must fulfill the values listed
in Table 1.2 at the receiver. These power levels are very weak and the
spectrum is spread, therefore they cannot be directly observed from a
spectrum analyzer. Even when the signal is amplified to a reasonable
power level, the spectrum of the C/A code cannot be observed because
the noise is stronger than the signal.
14 Fundamentals of Electrical Engineering
Table 1.3. Modulo-2 Addition
Input 1 Input 2 Output
0 0 0
0 1 1
1 0 1
1 1 0
The C/A and P (or Y) codes can be considered as the measuring rods
– they provide the means by which a GPS receiver can measure one-
way distances to the satellites. Both codes have the characteristics of
random noise, but are in fact binary codes generated by mathematical
algorithms and are therefore referred to as ”pseudo-random-noise” (or
PRN) codes. Figure 1.7 illustrates the C/A code generation procedure
based on ”Gold Codes”. Tapped Feedback Shift Registers (TFSR) are
used to generate a sequence of ”0”s and ”1”s (also known as maximum-
length sequence) at the clock rate of 1.023 MHz. At each clock pulse
the bits in the registers are shifted to the right where the contents of
the rightmost register is read as output. A new value in the leftmost
register is created by the modulo-2 addition (Table 1.3) of the contents
of a specified group of registers. In the case of the C/A code two 10-bit
TFSRs are used, each generating a Gold Code (See Figure 1.8): (1) the
G1 (represented here as the polynomial: 1 +X
3
+X
10
), and (2) the G2
(represented here as the polynomial: 1+X
2
+X
3
+X
6
+X
8
+X
9
+X
10
).
The output of the G1 TFSR (rightmost register) is modulo-2 added to
the register contents of the G2. Different combinations of the outputs
of the registers of G2 (or ”taps” from the register) when added to the
output of the G1 code lead to different PRN codes. There are 37 unique
codes that can be generated in such a straightforward manner. Among
these 37 codes, 32 are utilized for the C/A codes of 32 satellites, but
only 24 satellites are in orbit. The other five outputs are reserved for
other applications such as ground transmission.
Table 1.4 lists the code phase assignments. In this table there are five
columns and the first column gives the satellite ID number, which is from
1 to 32. The second column gives the PRN signal number; and it is from
1 to 37. It should be noted that the C/A codes of PRN signal numbers
34 and 37 are the same. The third column provides the code phase
selections that are used to form the output of the G2 generator. The
fourth column provides the code delay measured in chips. This delay is
the difference between the maximal length sequence (MLS) output and
the G2 output. This is redundant information of column 3, because once
Global Positioning System 15
Figure 1.7. C/A Code Generator (Reproduced from [11])
the code phase selections are chosen this delay is determined. The last
column provides the first 10 bits of the C /A code generated for each
satellite. These values can be used to check whether the generated code
is wrong. This number is in an octal format.
4.3 Correlation Properties of C/A-Code
The PRN sequences have two special properties. First, PRN se-
quences are nearly orthogonal to each other. The sum of the term-
by-term products of two different sequences is nearly zero. For satellites
k and l, which are assigned unique PRN sequences x
(k)
and x
(l)
, respec-
tively, have the following correlation:
1022

i=0
x
(k)
(i) · x
(l)
(i + n) ≈ 0, for all n, k = l, (1.3)
where x
(·)
(1023 + m) = x
(·)
(m). The left-hand side of (1.3) defines the
cross-correlation function of the two sequences for shift n. The PRN
16 Fundamentals of Electrical Engineering
Table 1.4. Code Phase Assignments
Satellite ID GPS PRN Code Phase Code Delay First 10 Chips
Number Signal Number Selection Chips C /A Octal
1 1 2 ⊕6 5 1440
2 2 3 ⊕7 6 1620
3 3 4 ⊕8 7 1710
4 4 5 ⊕9 8 1744
5 5 1 ⊕9 17 1133
6 6 2 ⊕10 18 1455
7 7 1 ⊕8 139 1131
8 8 2 ⊕9 140 1454
9 9 3 ⊕10 141 1626
10 10 2 ⊕3 251 1504
11 11 3 ⊕4 252 1642
12 12 5 ⊕6 254 1750
13 13 6 ⊕7 255 1764
14 14 7 ⊕8 256 1772
15 15 8 ⊕9 257 1775
16 16 9 ⊕10 258 1776
17 17 1 ⊕4 469 1156
18 18 2 ⊕5 470 1467
19 19 3 ⊕6 471 1633
20 20 4 ⊕7 472 1715
21 21 5 ⊕8 473 1746
22 22 6 ⊕9 474 1763
23 23 1 ⊕3 509 1063
24 24 4 ⊕6 512 1706
25 25 5 ⊕7 513 1743
26 26 6 ⊕8 514 1761
27 27 7 ⊕9 515 1770
28 28 8 ⊕10 516 1774
29 29 1 ⊕6 859 1127
30 30 2 ⊕7 860 1453
31 31 3 ⊕8 861 1625
32 32 4 ⊕9 862 1712
** 33 5 ⊕10 863 1745
** 34* 4 ⊕10 950 1713
** 35 1 ⊕7 947 1134
** 36 2 ⊕8 948 1456
** 37* 4 ⊕10 950 1713
*34 and 37 have the same C /A code.
**GPS satellites do not transmit these codes; they are reserved for other uses.
Global Positioning System 17
Figure 1.8. G1, G2 maximal-length sequence generators (Reproduced from [11])
sequences are nearly uncorrelated for all shifts.
The second important property is that a PRN sequence is nearly un-
correlated with itself, except for zero shift. For a C/A-code, the auto-
correlation function of a sequence for shift n is defined as:
1022

i=0
x
(k)
(i) · x
(k)
(i + n) ≈ 0, for all |n| ≥ 1. (1.4)
In order to detect a weak signal in the presence of strong signals, the
autocorrelation peak of the weak signal must be stronger than the cross-
correlation peaks from the strong signals. If the codes are orthogonal,
the cross correlations will be zero. However, the Gold codes are not
orthogonal but near orthogonal, implying that the cross correlations are
not zero but have small values.The cross correlation of the Gold code is
listed in Table 1.5.
For the C/A-code n= even = 10, thus, P = 1023. Using the relations
in the above table, the cross-correlation values are: -65/1023 (occurrence
18 Fundamentals of Electrical Engineering
Table 1.5. Cross-Correlation of Gold Codes
Number of Shift Normalized Cross Probability of
Code Period Register Stages Correlation Level Level
P = 2
n
−1 n=odd
_
¸
¸
¸
_
¸
¸
¸
_
−
2
(n+1)/2
+ 1
P
0.25
−
1
P
0.5
2
(n+1)/2
−1
P
0.24
P = 2
n
−1 n=even
_
¸
¸
¸
_
¸
¸
¸
_
−
2
(n+1)/2
+ 1
P
0.125
−
1
P
0.75
2
(n+1)/2
−1
P
0.125
12.5%), -1/1023 (75%), and 63/1023 (12.5%). The autocorrelation of the
C/A-codes of satellite 19 and the cross correlation of satellites 19 and 31
are shown in Figures 1.9(a) and 1.9(b) respectively. These satellites are
arbitrarily chosen. In Figure 5.6a, the maximum of the autocorrelation
peak is 1023, which equals the C/A-code length. The position of the
maximum peak is deliberately shifted to the center of the figure for a
clear view. The rest of the correlation has three values 63, -1, and -65.
The cross-correlation shown in Figure 1.9(b) also has three values 63,
-1, -65.
These are the values calculated by using equations in Table 1.5. The
difference between the maximum of the autocorrelation to the cross cor-
relation determines the processing gain of the signal. In order to generate
these figures, the outputs from the C /A code generator must be 1 and
-1, rather than 1 and 0.
4.4 GPS Navigation Message
GPS transmits navigation data message at 50 bits per second. The
C/A-code is a bi-phase coded signal which changes the carrier phase
between 0 and π at a rate of 1.023 MHz. The navigation data bit is also
bi-phase phase code, but its rate is only 50 Hz, or each data bit is 20 ms
long. Since the C/A-code is 1 ms, there are 20 C/A-codes in one data
bit. Thus, in one data bit all 20 C/A-codes have the same phase. The
Global Positioning System 19
Figure 1.9. Auto- and Cross-Correlation of C/A Code
message is formatted into frames of 1500 bits. It takes thirty seconds
to transmit a frame. Each frame is organized into five subframes. Each
subframe is six seconds long and contains ten 30-bit words (see Figure
1.10). Subframe 1-3 typically repeat the same information from frame to
frame. Subframes 4-5 of the consecutive frames, however, contain differ-
ent pages of the navigation message. It takes 25 frames (12.5 minutes) to
transmit the complete navigation message. A unit of 25 frames is called
a Master Frame. The information content of the various subframe is
summarized below.
Subframe 1: satellite clock corrections, health indicators, age of data
Subframe 2-3: satellite ephemeris parameters
Subframe 4: ionosphere model parameters, UTC data, almanac and
health status data for satellites numbered 25 and higher
Subframe 5: almanac and health status data for satellites numbered
1-24
20 Fundamentals of Electrical Engineering
Figure 1.10. GPS Navigation Message Organization
The first two words of each six-second subframe have a spatial signif-
icance: Telemetry word (TLM) and Hand-over word (HOW). The TLM
contains a fixed 8-bit synchronization pattern and a 14-bit message. The
HOW word provides time information (seconds into the week) which al-
lows a receiver to acquire the week-long P(Y)-code segment.
5. GPS Observables
GPS provides two types of measurements. Code tracking provides
estimates of instantaneous ranges to the satellites. The code phase mea-
surements at an instant from different satellites have a common bias
and are, therefore, called pseudoranges. Carrier phase tracking provides
measurements of the reveived carrier phase relative to the phase of a si-
nusoidal signal generated by the receiver clock. The carrier phase gives
a precise measurement of change in the satellite-user pseudorange over
a time interval, and estimate of its instantaneous rate, or Doppler fre-
Global Positioning System 21
quency.
5.1 Measurement Models - Code Phase
Measurements
A code phase measurement made by a GPS receiver can be obtained
from the transit time of the signal from a satellite to the receiver. The
signal transit time is defined as the time difference between the time of
reception, as determined by the receiver clock, and the time of trans-
mission, as marked on the signal by the satellire. In a GPS receiver, the
transit time is measured as the amount of time shift required to align
the C/A-code replica generated at the receiver with the signal received
from the satellite. There is no way that the satellite and the receiver
clocks can be perfectly synchronized, because each of them keeps time
independently (Each satellite generates its signals in accordance with a
clock on board, while the receiver generates a replica of each signal in
accordance with its own clock). Therefore the code phase measurements
are biased. The corresponding biased range, or pseudorange, is defined
as the transit time so measured multiplied by the speed of light in a
vacuum.
There are three different time scales to deal with: two of these are
the times kept by the satellite and receiver clocks, respectively. The
third is a common time reference. Usually it is GPS time. GPS time
is a composite time scale derived from the times kept by clocks at GPS
monitor stations.
Let τ be the transit time of the signal from a satellite arriving at the
receiver at time t per GPS time. Let t
s
(t−τ) be the time of transmission
stamped on the signal, where the superscript ”s” indicates it was derived
from the satellite’s clock. Let t
u
(t) be the arrival time measured by the
user’s receiver clock. The measured pseudorange is determined from the
apparent transit time as (See Figure 1.11:
ρ(t) = c[t
u
(t) −t
s
(t −τ)]. (1.5)
In equation (1.5), t and τ are unknow parameters that are to be
estimated by the receiver’s signal processing algorithm. To simplify no-
tation, we shall make no reference to the satellite ID or carrier frequency,
when dealing with equation (1.5).
22 Fundamentals of Electrical Engineering
Figure 1.11. A Conceptual View of Pseudorange Measurements
The time scales of the receiver and the satellite clocks can be related
to GPS time as follows.
t
u
(t) = t + δt
u
(t), (1.6)
t
s
(t −τ) = (t −τ) + δt
s
(t −τ), (1.7)
where δt
u
is the receiver clock bias and δt
s
is the satellite clock bias. Here
both δt
u
and δt
s
are measured relative to GPS time. In our notation,
δt
u
and δt
s
reflect the amounts by which the receiver and satellite clocks
are advanced in relation to GPS time. The satellite clock bias term is
maintained by the GPS Control Segment. The clock bias is broadcast
in the navigation message in terms of the coefficient of a quadratic poly-
nomial in time.
By substituting equations (1.6) and (1.7) into the measured pseudor-
ange (1.5), we can obtain the following equation:
ρ(t) = c[t + δt
u
(t) −(t −τ + δt
s
(t −τ))] + ε
ρ
(t)
= cτ + c[δt
u
(t) −δt
s
(t −τ)] + ε
ρ
(t),
(1.8)
Global Positioning System 23
where ε
ρ
denotes the unmodelled effects, such as modelling error, mea-
surement error, etc., in the code phase measurement. The transit time
multiplied by the speed of light in a vacuum can be modeled as:
cτ = r(t, t −τ) + I
ρ
(t) + T
ρ
(t), (1.9)
where r(t, t − τ) is the geometric range (or true range) between the
user position at time t and the satellite position at (t − τ). I
ρ
and
T
ρ
represent the delays associated with the transmission of the signal
through the ionosphere and the troposphere, respectively. Both of them
are positive values. Upon substitution of equation (1.9) into equation
(1.8), we then obtain the following equation.
ρ = r + c[δt
u
−δt
s
] + I
ρ
+ T
ρ
+ ε. (1.10)
Here, for simplicity, we have dropped explicit reference to the measure-
ment epoch t. Equation (1.10) is the basic measurement equation for
computing the position velocity,and time of the receiver. Ideally, one
would have liked to measure the geometric range r from the satellite to
the receiver. In practice, however, we can only have the pseudorange
ρ instead. How accurate an estimate of position, velocity, or time we
can obtain from these measurements would depend upon our ability to
compensate for, or eliminate, the biases and errors.
The radius of the earth is 6,378 km around the equator and 6,357
km passing through the poles, and the average radius can be considered
as 6,368 km. The radius of the satellite orbit is 26,560 km, which is
about 20,192 km (26,560 - 6,368) above the earths surface. This height
is approximately the shortest distance between a user on the surface of
the earth and the satellite, which occurs at around zenith or an elevation
angle of approximately 90 degrees. Most GPS receivers are designed to
receive signals from satellites above 5 degrees. For simplicity, let us
assume that the receiver can receive signals from satellites at the zero-
degree point. The distance from a satellite on the horizon to the user is
25,785 km (
_
26, 5602
2
−6, 3682
2
). These distances are shown in Figure
1.12.
From the distances in Figure 1.12) one can see that the time delays
from the satellites are in the range of 67 ms to 86 ms. If the user is on the
surface of the earth, the maximum differential delay time from two dif-
ferent satellites should be within 19 (86V67) ms. The C/A-code repeats
each millisecond, and the code correlation process essentially provides
24 Fundamentals of Electrical Engineering
Figure 1.12. Earth and Circular Satellite Orbit
a measurement of pesudo-transit time modulo 1 ms. The measurement
is ambigious in whole milliseconds. This ambiguity, however, is easily
resolved if the user has a rough idea of his location within hundreds of
kilometers. The week-long P(Y)-code provides unambiguous pseudor-
anges.
5.2 Measurement Models - Carrier Phase
Measurements
A measurement much more precise than that of code phase is the
phase of the carrier received from a satellite. The carrier phase mea-
surement is the difference between the phases of the receiver-generated
carrier signal and the carrier received from a satellite at the instant of
the measurement.
Global Positioning System 25
Consider first an idealized case of error-free measurements with per-
fect and synchronized satellite and receiver clocks, and no relative mo-
tion between th satellite and the user. In this model, the carrier phase
measurement would remain fixed at a fraction of a cycle, and the dis-
tance between a satellite and the receiver would be an unknown number
of whole cycle plus the measured fractional cycle. The measurement,
however, contains no information regarding the number of whole cycles,
referred to as the integer ambiguity.
Now suppose that the carrier phase is tracked while the receiver or
the satellite moves so that the distance between them grows by a wave-
length. The corresponding carrier phase measurement would be a full
cycle plus the fractional cycle phase measured the movement began. To
measure carrier phase in GPS, a receiver acquires phase lock with the
satellite signal, measures the initial fractional phase difference between
the received and receiver-generated signals, and from then on tracks the
change in this measurement, counting full carrier cycles and keeping
track of the fractional cycle at each epoch.
In the absence of clock biases and measurement errors, the carrier
phase measurement in units of cycles can be written as
φ(t) = φ
u
(t) −φ
s
(t −τ) + N, (1.11)
where φ
u
(t) is the phase of the receiver-generated signal; φ
s
(t −τ) is
the phase of the signal received from the satellite at time t, or the phase
of the signal at the satellite at time (t −τ); τ is the transit time of the
signal; and N is the integer ambiguity. Estimation of N is referred to
as integer ambiguity resolution or intialization. Simplifying the above
expression by writing phase as the product of frequency and time gives
φ(t) = f · τ + N =
r(t, t −τ)
λ
+ N, (1.12)
where f and λ are the carrier frequency and wavelength, respectively,
and r(t, t −τ), as before, is the geometric range between the user posi-
tion at time t and the satellite position at (t −τ).
Accounting for the various measurement errors, clcok biases and initial
phase offsets, equation (1.12) can be rewritten as
φ =
1
λ
[r + I
φ
+ T
φ
] +
c
λ
(δt
u
−δt
s
) + N + ε
φ
, (1.13)
26 Fundamentals of Electrical Engineering
where I
φ
and T
φ
are the ionospheric and tropospheric propagation de-
lays in meters, respectively. Here, again for simplicity, we have dropped
explicit reference to the measurement epoch t. Note that the carrier
phase measurement in equation (1.13) is in units of cycles.
The change in carrier phase measurement over a time interval cor-
responds to change in both the user-satellite range and receiver clock
bias. This change is usually referred to as integrated Doppler or delta
pseudorange. The rate of change of the carrier phase measurement gives
the pseudorange rate, which is made up of the actual range rate plus the
receiver clock frequency bias.
The appearance of equation (1.13) is pretty much similar to equation
(1.10) which is derived based on the code phase measurement. Compar-
ing (1.13) and (1.10), it can be seen that both the carrier and code phase
measurements are contaminated by the same error sources. But there is
an important difference: Code tracking provides essentially unambiguous
pseudorange, while the carrier phase measurements being encumbered
with integer ambiguity. On the other hand, code phase measurements
are coarse in comparison with the carrier phase measurements. Luckily
enough, the integer ambiguity remain constant as long as the carrier
tracking loop remains lock. Any break in tracking could change the in-
teger values.
In order to take full advantage of the precision of the carrier phase
measurements to obtain accurate position estimates, we have to resolve
the integer ambiguities and compensate for the various errors. With
only equation (1.13) in hand, there is no way that we can expect to
determine the value of integer N. We will have to look for alternative
approaches and formulations to take advantage of the precision of these
measurements.
When phase measurement for only one frequency (L1 or L2) are avail-
able, the most direct approach is as follows. The measurements are
modeled by equation (1.13), and the linearized equations are processed.
Depending on the model chosen, a number of unkonws is estimated along
with N in a common adjustment. In this geometric approach, the un-
modeled errors affect all estimated parameters. Therefore, the integer
nature of the ambiguities is lost and they are estimated as real values.
To fix ambiguities to integer values, a sequential adjustment could be
performed. After an initial adjustment, the ambiguity with a computed
Global Positioning System 27
value closest to an integer and with minimum standard error is consid-
ered to be determined most reliably. This bias is then fixed, and the
adjustment is repeated (with one less unknown) to fix another ambi-
guity and so on. When using double-differences over short baselines,
this approach is usually successful. The critical factor is the ionospheric
refraction which must be modeled and which may prevent a correct reso-
lution of all ambiguities. Detailed discussions of the ambiguity resolution
as well as the carrier phase positioning are beyond the scope of this book,
hence are omitted.
6. PVT Estimation
In this section, we consider estimation of position, velocity, and time
(PVT) based on measurements of pseudoranges. With such measure-
ments available from four or more satellites, PVT can be estimated in-
stantaneously.
The pseudorange measurement from the k-th satellite at epoch t (GPS
time) can be modeled as
ρ
(k)
= r
(k)
(t, t −τ) + c[δt
u
(t) −δt
(k)
(t −τ)] + I
(k)
(t) + T
(k)
(t) + ε
(k)
(t),
(1.14)
where k = 1, 2, ....K denotes the k-th satellite. A user would correct
each measured pseudorange for the known errors using parameter val-
ues in the navigation message from the satellite. The main corrections
available to an SPS user are: (1) satellite clock offset relative to GPS
time, (2) relativity effect, and (3) ionospheric delay using the parameter
values for the Klobuchar model[6]. Step-by-step implementation of these
corrections is given in the GPS Interface Control Document[4].
Denote by ρ
(k)
c
the pseudorange obtained after accounting for the
satellite clock offset and compensating for the remaining errors in the
measurements to the extent it is practical for a user. Equation (1.14)
can be rewritten for the corrected pseudorange measurements as follows.
ρ
(k)
c
= r
(k)
+ cδt
u
+ ˜ ε
(k)
ρ
, (1.15)
28 Fundamentals of Electrical Engineering
wherer the term ˜ ε
(k)
ρ
denotes the combined effect of the residual errors.
Let x = (x, y, z) be the position of the user at the time of measurement.
Let x
(k)
= (x
(k)
, y
(k)
, z
(k)
), k = 1, 2, ..., K, represents the position of the
k-th satellite at the time of signal transmission. The user-to-satellite
geometric range can be represented as (See Figure 1.13):
r
(k)
=
_
_
x
(k)
−x
_
2
+
_
y
(k)
−y
_
2
+
_
z
(k)
−z
_
2
= x
(k)
−x.
Substituting the above equation into (1.15) gives
ρ
(k)
c
= x
(k)
−x + b + ˜ ε
(k)
ρ
, (1.16)
where b := cδt
u
with units of meters.
Figure 1.13. The ranges to GPS satellites measured by a receiver have a common
bias and are called pseudorange.
Usually the user clock error cannot be corrected through received in-
formation. Thus, it will remain as an unknown. A simple approach to
solving the K equations (1.16) is to linearized them about an approxi-
mate user position, and solve iteratively. The idea is to start with rough
Global Positioning System 29
estimates of the user position and clock bias, and refine them in stages
so that the estimates fit the measurements better. The approach de-
scribed below is generally referred to as Newton-Raphson method. Let
x
0
= (x
0
, y
0
, z
0
) and b
0
be the first guesses of the user position and
receiver clock bias, respectively. From (1.16), the corrected pseudor-
ange measurement from satellite k is ρ
(k)
c
. Let ρ
(k)
0
be the corresponding
approximation based on the initial guesses x
0
and b
0
. Then
ρ
(k)
0
= x
(k)
−x
0
+ b
0
. (1.17)
Let the true position and true clock bias be represented as x = x
0
+δx
and b = b
0
+ δb, where δx and δb are the unknown corrections to be
applied to our initial estimates. Equation (1.16) has a first-order Taylor
series approximation in (δx, δb) of the following form:
δρ
(k)
= ρ
(k)
c
−ρ
(k)
0
= x
(k)
−x
0
−δx −x
(k)
−x
0
+ (b −b
0
) + ˜ ε
(k)
ρ
≈ −
(x
(k)
−x
0
)
x
(k)
−x
0

· δx + δb + ˜ ε
(k)
ρ
= −1
(k)
· δx + δb + ˜ ε
(k)
ρ
,
(1.18)
where 1
(k)
is the estimated line-of-sight unit vector directed from the
initial estimate of the user position to satellite k. The elements of 1
(k)
are direction cosines of the vector drawn from the estimated receiver
location to the satellite.
1
(k)
=
1
x
(k)
−x
0

_
x
(k)
−x
0
, y
(k)
−y
0
, z
(k)
−z
0
_
T
(1.19)
It wolud be more easily to derive (1.18) if we use the long-hand notaton
as
_
_
x
(k)
−x
0
−δx
_
2
+
_
y
(k)
−y
0
−δy
_
2
+
_
z
(k)
−z
0
−δz
_
2
≈
_
_
x
(k)
−x
0
_
2
+
_
y
(k)
−y
0
_
2
+
_
z
(k)
−z
0
_
2
−
_
x
(k)
−x
0
_
δx +
_
y
(k)
−y
0
_
δy +
_
z
(k)
−z
0
_
δz
_
_
x
(k)
−x
0
_
2
+
_
y
(k)
−y
0
_
2
+
_
z
(k)
−z
0
_
2
.
A more compact form of the above equation is written as:
x
(k)
−x
0
−δx ≈ x
(k)
−x
0
−
(x
(k)
−x
0
)
x
(k)
−x
0

· δx.
30 Fundamentals of Electrical Engineering
The set of K linear equations (1.18) can be written in matrix notation
as
δρ =
_
¸
¸
¸
¸
_
δρ
(1)
δρ
(2)
.
.
.
δρ
(K)
_
¸
¸
¸
¸
_
=
_
¸
¸
¸
¸
_
(−1
(1)
)
T
1
(−1
(2)
)
T
1
.
.
.
.
.
.
(−1
(K)
)
T
1
_
¸
¸
¸
¸
_
_
δx
δb
_
+ ˜ ε
ρ
. (1.20)
These are K (≥ 4) linear equations with four unknown: (δx, δy, δz) and
δb. Equation (1.20) can be written in a more compact matrix form:
δρ = G
_
δx
δb
_
˜ ε
ρ
, (1.21)
where
G :=
_
¸
¸
¸
¸
_
(−1
(1)
)
T
1
(−1
(2)
)
T
1
.
.
.
.
.
.
(−1
(K)
)
T
1
_
¸
¸
¸
¸
_
is a (K×4) matrix characterizing the user-satellite geometry. Therefore
G is usually referred to as the geometry matrix.
If K = 4 we can generally solve the four equations for four unknown
directly. A problem would arise if the equations are linearly dependent,
making G rank-deficient. This could happen, for example, if the eleva-
tion angles of all satellites measured from the user position are the same.
In this case, the tips of the unit line-of-sight vectors would all lie in a
plane, and rank(G)=3.
If K > 4, we have an over-determined system of equations, and G
would be full-rank, in general. We can now look for a solution which fits
the measurement best. It is common to use the criterion of least-squares:
The best solution is the one for which the sum of squared residuals is
smallest. The least-squares solution for the corrections to the initial
estimates can be written as:
_
δˆ x
δ
ˆ
b
_
= (G
T
G)
−1
G
T
δρ, (1.22)
and the new, improved estimates of the user position and clock bias are
Global Positioning System 31
ˆ x = x
0
+ δ˜ x,
ˆ
b = b
0
+ δ
ˆ
b.
(1.23)
The observation equations may now be linearized about these new es-
timates of the user position and clock bias, and the solution may be
iterated until the change in the estimates is sufficiently small.
The above approach can be summarized in the following steps:
1 Choose a nominal position and user clock bias x
0
, y
0
, z
0
, b
0
to
represent the initial condition. For example, the position can be
the center of the earth and the clock bias zero. In other words, all
initial values are set to zero.
2 Use equation (1.17) to calculate the pseudorange ρ
0
These ρ
0
values
will be different from the measured values. The difference between
the measured values and the calculated values is δρ.
3 Use the calculated ρ
0
in equation (1.19) to calculate the unit line-
of-sight vectors.
4 Use equation (1.22) to find δx, δy, δz, and δb.
5 From the absolute values of δx, δy, δz, δb calculate
δv :=
_
δx
2
+ δy
2
+ δz
2
+ δb
2
.
6 Compare δv with an arbitrary chosen threshold; if δv is greater
than the threshold, the following steps will be needed.
7 Add these values δx, δy, δz, δb to the initial chosen position x
0
,
y
0
, z
0
, and the clock bias b
0
; a new set of position and clock bias
can be obtained and they will be expressed as in equation (1.23).
These values will be used as the initial position and clock bias in
the following calculations.
8 Repeat the procedure from 1 to 7, until δv is less than the thresh-
old. The final solution can be considered as the desired user posi-
tion and clock bias.
32 Fundamentals of Electrical Engineering
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[2] FRP, 1999 Federal Radionavigation Plan, U.S. Department of De-
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