Unit III GPS Receivers and Errors

GPS
UNIT III
GPS RECEIVERS & DATA ERRORS
-By
GVNSK Sravya
Assistant Professor
ECE Dept.
GNITS

CONTENTS
• Receiver Architecture
• Receiver Design Options,
• Antenna Design,
• GPS Error Sources,
• SA Errors,
• Propagation Errors,
• Ionospheric Error,
• Tropospheric Error,
• Multipath Errors,
• Estimation using dual frequency GPS
receiver,
• Methods of Multipath Mitigation,
• Ephemeris Data Errors,
• Clock Errors

Receiver Architecture
• Most of the GPS receivers, perform certain basic functions with
variations in their design.
• The basic functions appears as a block in the GPS Receiver
Architecture.
• The architecture of GPS receiver design is shown in the below figure.

GPS RECEIVER ARCHITECTURE
GPS Receiver Architecture

RADIO FREQUENCY STAGES (FRONT END)
• The purpose of the receiver front end is to filter and amplify the
incoming GPS signal.
• The GPS signal power available at the receiver antenna output
terminals is extremely small and can easily be masked by interference
from more powerful signals adjacent to the GPS passband.

CONTD…
• Therefore, RF amplification in the receiver front end provides as
much as 35–55 dB of gain.
• Usually the front end will also contain passband filters to reduce
out-of-band interference without degradation of the GPS signal
waveform.

CONTD…
• The bandwidth of both the L1 and L2 GPS signals is 20 MHz (±10
MHz on each side of the carrier), and sharp cutoff bandpass filters
are required for out-of-band signal rejection.
• However, the small ratio of passband width to carrier frequency
makes the design of such filters infeasible.

CONTD…
• Consequently, filters with wider skirts are commonly used as a
first stage of filtering, which also helps prevent front-end
overloading by strong interference, and the sharp-cut off filters
are used later after down conversion to intermediate frequencies
(IFs).

FREQUENCY DOWN CONVERSION
AND IF AMPLIFICATION
• After amplification in the receiver front end, the GPS signal is
converted to a lower frequency called an intermediate frequency
for further amplification and filtering.
• Down conversion is accomplished by multiplying the GPS signal
by a sinusoid called the local-oscillator signal in a mixer.

CONTD…
• The local-oscillator frequency is either larger or smaller than
the GPS carrier frequency by an amount equal to the IF.
• Sum frequency components are also produced, but these are
eliminated by a simple bandpass filter following the mixer.

SIGNAL-TO-NOISE RATIO
• The noise power in IF bandwidth is given by
N = kTeB
Where,
k = 1.3806 × 10−23 J/K,
B is the bandwidth in Hz, and
Te is the effective noise temperature in degrees Kelvin.

CONTD…
• A typical effective noise temperature for a GPS receiver is 513 K,
resulting in a noise power of about −138.5 dBW in a 2-MHz bandwidth
and −128.5 dBW in a 20-MHz bandwidth.
• Using −154.6 dBW for the received signal power obtained at the receiver,
the SNR in a 20-MHz bandwidth is seen to be −154.6 −(−128.5) = −26.1
dB.

CONTD…
• Consequently the SNR in a 2-MHz bandwidth is (−154.6 − 0.5)
− (−138.5) = −16.6 dB. In either case it is evident that the
signal is completely masked by noise.

DIGITIZATION
• In modern GPS receivers digital signal processing is used to track
the GPS signal, make pseudo range and Doppler measurements, and
demodulate the 50-bps (bits per second) data stream.
• For this purpose the IF signal is sampled and digitized by an analog-
to-digital converter (ADC).

BASEBAND SIGNAL PROCESSING
• Baseband signal processing refers to acquiring and tracking the GPS
signal, extract the 50- bps (bits per second) navigation data, and
provide measurements of code and carrier pseudo ranges and Doppler.

CARRIER TRACKING
• Tracking of the carrier phase and frequency is accomplished by
using feedback control of a numerically controlled oscillator (NCO)
to frequency shift the signal to precisely zero frequency and phase.

CODE TRACKING AND SIGNAL
SPECTRAL DESPREADING
• The digitized IF signal, which has a wide bandwidth due to the C/A or
P code modulation, is completely obscured by noise.
• The signal power is raised above the noise power by despreading, in
which the digitized IF signal is multiplied by a receiver-generated
replica of the code precisely time-aligned with the code on the
received signal.

CONTD…
• The despreading process removes the code from the signal, thus
concentrating the full signal power into the approximately 50-Hz
baseband bandwidth of the data modulation.
• Subsequent filtering can be employed to dramatically raise the SNR to
values permitting observation and measurement of the signal.

CONTD…
• As an example, in a GPS receiver a typical SNR in a 2-MHz IF bandwidth is
−16.6 dB.
• After despreading and 50-Hz lowpass filtering the total signal power is still
about the same, but the bandwidth of the noise has been reduced from 2 MHz to
about 50 Hz, which increases the SNR by the ratio 2 × 106/50, or 46 dB.
• The resulting SNR is therefore −16.6 + 46.0 = 29.4 dB.

RECEIVER DESIGN CHOICES
• Number of Channels and Sequencing Rate
 Single channel receivers
 2 channel receivers
 Receivers with 3-5 channels
 Multi channel all in view receivers
• L2 Capability
• Dual-Frequency Ionospheric Correction
• Improved Carrier Phase Ambiguity Resolution in High-Accuracy Differential Positioning

CONTD…
• Code Selections: C/A, P, or Codeless
• Access to SA Signals
• Differential Capability
• Errors Common to Both Receivers
• Aiding Inputs
• INS Aiding
• Aiding with additional navigation inputs
• Altimeter aiding
• Clock aiding

ANTENNA DESIGN
• Right hand circularly polarized antennas are most commonly used GPS
antennas to match the incoming signal and the spatial reception pattern is a
hemisphere.
• Such a pattern permits reception of signals from satellites in any azimuthal
direction from zenith down to the horizon.

VARIOUS TYPES OF GPS ANTENNAS
• Patch antennas
• Dome antennas
• Helical antennas
• Choker ring antennas
• Phased array antennas
• Blade antennas

GPS ERRORS
• GPS pseudo range and carrier-phase measurements are both affected by
several types of random errors and systematic errors which will affects the
accuracy of the measurement.

GPS ERROR SOURCES
• Errors arise from a variety of sources.
1. Satellite positions (geometry)
2. Weather
3. Multipath
4. Timing errors

GPS ERRORS
• These errors may be classified as
1. those originating at the satellites
2. those originating at the receiver
3. those that are due to signal propagation

SELECTIVE AVAILABILITY ERRORS
• Selective availability (SA) is a technique to deny accurate real-time
autonomous positioning to unauthorized users.
• SA was officially activated on March 25, 1990 on Block II GPS satellites
to deny the accurate positioning.

• SA introduces two types of errors.
i. Delta error (δ) - results from dithering the satellite clock.
ii. Epsilon error (ε) - is an additional slowly varying orbital error.
• SA turned on, nominal horizontal and vertical errors could be up to 100m
and 156m, respectively.

• U.S. government discontinued SA on May 1, 2000, resulting in a much-
improved GPS accuracy.
• With the SA turned off, the nominal GPS horizontal and vertical accuracies
would be in the order of 22m and 33m.

EPHEMERIS
DATA
ERRORS
• Ephemeris errors
are also called as
orbital errors.

EPHEMERIS DATA ERRORS
• Satellite positions are a function of time.
• Forces acting on the GPS satellites are not perfect.
• Errors in the estimated satellite positions known as ephemeris errors.
• Satellite ephemerides are determined by the master control station of the
GPS ground segment based on monitoring of individual signals by four
monitoring stations.

• Because the locations of these stations are known precisely, an “inverted”
positioning process can calculate the orbital parameters of the satellites as
if they were users.
• This process is aided by precision clocks at the monitoring stations and by
tracking over long periods of time with optimal filter processing.

• Based on the orbital parameter estimates thus obtained, the master control
station uploads the ephemeris data to each satellite, which then transmits
the data to users via the navigation data message.
• Errors in satellite position when calculated from the ephemeris data
typically result in range errors less than 1 m.
• Improvements in satellite tracking will undoubtedly reduce this error
further.

SATELLITE AND RECEIVER CLOCK ERRORS
• Each GPS Block II and Block IIA satellite contains four atomic clocks,
two cesium and two rubidium.
• The newer generation Block IIR satellites carry rubidium clocks only. One
of the onboard clocks, primarily a cesium for Block II and IIA, is selected
to provide the frequency and the timing requirements for generating the
GPS signals.
• The GPS satellite clocks, although highly accurate, are not perfect.

• Satellite clock error is about 8.64 to 17.28 ns per day.
• The corresponding range error is 2.59m to 5.18m, which can be easily
calculated by multiplying the clock error by the speed of light. (One
nanosecond error is equivalent to a range error of about 30 cm).

• Satellite clock errors cause additional errors to the GPS measurements.
• These errors are common to all users observing the same satellite and can
be removed through differencing between the receivers.
• Applying the satellite clock correction in the navigation message can also
correct the satellite clock errors.

• GPS receivers, in contrast, use inexpensive crystal clocks, which are much
less accurate than the satellite clocks.
• As such, the receiver clock error is much larger than that of the GPS
satellite clock. It can, however, be removed through differencing between
the satellites or it can be treated as an additional unknown parameter in the
estimation process.

• Precise external clocks (usually cesium or rubidium) are used in some
applications instead of the internal receiver clock.
• Although the external atomic clocks have superior performance compared
with the internal receiver clocks, they cost between a few thousand dollars
for the rubidium clocks to about $20,000 for the cesium clocks.

TROPOSPHERIC PROPAGATION ERRORS
• The lower part of the earth’s atmosphere is composed of dry gases and
water vapor, which lengthen the propagation path due to refraction.
• The magnitude of the resulting signal delay depends on the refractive index
of the air along the propagation path and typically varies from about 2.5 m
in the zenith direction to 10–15 m at low satellite elevation angles.
• The troposphere is non dispersive at the GPS frequencies, so that delay is
not frequency dependent.

• In contrast to the ionosphere, tropospheric path delay is consequently the same
for code and carrier signal components.
• Therefore, this delay cannot be measured by utilizing both L1 and L2
pseudorange measurements, and either models and/or differential positioning
must be used to reduce the error.
• The refractive index of the troposphere consists of that due to the dry-gas
component and the water vapor component, which respectively contribute about
90% and 10% of the total.

• Knowledge of the temperature, pressure, and humidity along the
propagation path can determine the refractivity profile, but such
measurements are seldom available to the user.
• However, using standard atmospheric models for dry delay permits
determination of the zenith delay to within about 0.5 m and with an error at
other elevation angles that approximately equals the zenith error times the
cosecant of the elevation angle.

• These standard atmospheric models are based on the laws of ideal gases
and assume spherical layers of constant refractivity with no temporal
variation and an effective atmospheric height of about 40 km.
• Estimation of dry delay can be improved considerably if surface pressure
and temperature measurements are available, bringing the residual error
down to within 2–5% of the total.

• The component of tropospheric delay due to water vapor (at altitudes up to
about 12 km) is much more difficult to model, because there is
considerable spatial and temporal variation of water vapor in the
atmosphere.
• Fortunately, the wet delay is only about 10% of the total, with values of 5–
30 cm in continental midlatitudes

• Despite its variability, an exponential vertical profile model can reduce it to
within about 2–5 cm.
• In practice, a model of the standard atmosphere at the antenna location
would be used to estimate the combined zenith delay due to both wet and
dry components.
• Such models use inputs such as the day of the year and the latitude and
altitude of the user.

• The delay is modeled as the zenith delay multiplied by a factor that is a
function of the satellite elevation angle.
• At zenith, this factor is unity, and it increases with decreasing elevation
angle as the length of the propagation path through the troposphere
increases.
• Typical values of the multiplication factor are 2 at 30◦ elevation angle, 4 at
15◦, 6 at 10◦, and 10 at 5◦.

• The accuracy of the model decreases at low elevation angles, with decimeter level
errors at zenith and about 1 m at 10◦ elevation.
• Although a GPS receiver cannot measure pseudorange error due to the
troposphere, differential operation can usually reduce the error to small values by
taking advantage of the high spatial correlation of tropospheric errors at two
points within 100–200 km on the earth’s surface.
• However, exceptions often occur when storm fronts pass between the receivers,
causing large gradients in temperature, pressure, and humidity.

IONOSPHERIC
PROPAGATION ERRORS
• The uppermost part of the earth’s
atmosphere (50 km and 1000 km),
ultraviolet and X-ray radiations
coming from the sun interact with
the gas molecules and atoms.

IONOSPHERIC PROPAGATION ERRORS
• These interactions result in gas ionization, a large number of free,
negatively charged, electrons and positively charged, atoms and molecules,
such a region of the atmosphere where gas ionization takes place is called
the ionosphere.

• The electron density within the ionospheric region is not constant, it
changes with altitude. As such, the ionospheric region is divided into sub
regions, or layers, according to the electron density.
• The altitude and thickness of those layers vary with time, as a result of the
changes in the sun’s radiation and the Earth’s magnetic field.

• Generally, ionospheric delay is of the order of 5m to 15m, but can reach over 150m under
extreme solar activities, at midday, and near the horizon.
• The ionosphere is a dispersive medium, which means it bends the GPS radio signal and
changes its speed as it passes through the various Ionospheric layers to reach a GPS
receiver.
• Bending the GPS signal path causes a negligible error, particularly if the satellite elevation
angle is greater than 5°.

• It is the change in the propagation speed that causes a significant range
error.
• As the ionosphere is a dispersive medium, it causes a delay that is
frequency dependent.
• The lower the frequency, the greater the delay.

• The ionospheric delay is proportional to the number of free electrons along the GPS signal
path, called the total electron content (TEC).
• TEC, however, depends on a number of factors:
(1) the time of day (electron density level reaches a daily maximum in early afternoon and a
minimum around midnight at local time);
(2) the time of year (electron density levels are higher in winter than in summer);
(3) the 11-year solar cycle (electron density levels reach a maximum value approx. every 11
years.
(4) the geographic location (electron density levels are minimum in mid latitude regions and
highly irregular in polar, auroral, and equatorial regions).

• A particular location within the ionosphere is alternately illuminated by the sun
and shadowed from the sun by the earth in a daily cycle; consequently, the
characteristics of the ionosphere exhibit a diurnal variation in which the
ionization is usually maximum late in mid afternoon and minimum a few hours
after midnight.

• A curious fact is that the signal modulation (the code and data stream) is
delayed, while the carrier phase is advanced by the same amount.
• Thus, the measured pseudorange using the code is larger than the correct value,
while that using the carrier phase is equally smaller.
• The magnitude of either error is directly proportional to the total electron
content (TEC) in a tube of 1 m2 cross section along the propagation path.

• The TEC varies spatially, due to spatial nonhomogeneity of the ionosphere.
• Temporal variations are caused not only by ionospheric dynamics but also by
rapid changes in the propagation path due to satellite motion.
• The path delay for a satellite at zenith typically varies from about 1 m at night
to 5–15 m during late afternoon.

• At low elevation angles the propagation path through the ionosphere is much
longer, so the corresponding delays can increase to several meters at night and
as much as 50 m during the day.
• Since ionospheric error is usually greater at low elevation angles, the impact of
these errors could be reduced by not using measurements from satellites below
a certain elevation mask angle.

• However, in difficult signal environments, including blockage of some
satellites by obstacles, the user may be forced to use low elevation satellites.
• Mask angles of 5◦–7.5◦ offer a good compromise between the loss of
measurements and the likelihood of large ionospheric errors.

• The L1-only receivers in non differential operation can reduce ionospheric
pseudorange error by using a model of the ionosphere broadcast by the
satellites, which reduces the uncompensated ionospheric delay by about 50%
on the average.

• During the day errors as large as 10 m at mid latitudes can still exist after
compensation with this model and can be much worse with increased solar
activity.
• Other recently developed models offer somewhat better performance.

• However, they still do not handle adequately the daily variability of the TEC,
which can depart from the modeled value by 25% or more.
• The L1/L2 receivers in non differential operation can take advantage of the
dependence of delay on frequency to remove most of the ionospheric error.
• A relatively simple analysis shows that the group delay varies inversely as the
square of the carrier frequency.

• This can be seen from the following model of the code pseudorange
measurements at the L1 and L2 frequencies:

• The two equations can be solved for both ρ and k.

• With differential operation ionospheric errors can be nearly eliminated in many
applications, because ionospheric errors tend to be highly correlated when the
base and roving stations are in sufficiently close proximity.
• With two L1-only receivers separated by 25 km, the unmodeled differential
ionospheric error is typically at the 10–20-cm level.

• At 100 km separation this can increase to as much as a meter. Additional error
reduction using an ionospheric model can further reduce these errors by 25–
50%.

RECEIVER NOISE ERROR
• The receiver measurement noise results from the limitations of the receiver’s
electronics. A good GPS system should have a minimum noise level.
• The contribution of the receiver measurement noise to the range error will
depend very much on the quality of the GPS receiver.
• Typical average value for range error due to the receiver measurement noise is
of the order of 0.6m.

ANTENNAS PHASE CENTER VARIATION
• GPS antenna receives the incoming satellite signal and then converts its energy
into an electric current, which can be handled by the GPS receiver.
• The point at which the GPS signal is received is called the antenna phase
center.

• Generally, the antenna phase center does not coincide with the physical
(geometrical) center of the antenna.
• It varies depending on the elevation and the azimuth of the GPS satellite as
well as the intensity of the observed signal. As a result, additional range error
can be expected.

• The size of the error caused by the antenna phase-center variation depends on
the antenna type and is typically in the order of a few centimeters.
• It is, however, difficult to model the antenna phase-center variation and,
therefore, care has to be taken when selecting the antenna type.

• Mixing different types of antennas or using different orientations will not
cancel the error. Due to its rather small size, this error is neglected in most of
the practical GPS applications.

MULTIPATH ERROR
• A signal that bounces of a smooth object and hits the receiver antenna.
• Increases the length of time for a signal to reach the receiver.
• A big position error results
i. Gravel roads
ii. Open water
iii. Snow fields
iv. Rock walls
v. Buildings

MULTIPATH ERROR
• Technically speaking, multipath error occurs when the GPS signal arrives at
the receiver antenna through different paths.
• These paths can be the direct line of sight signal and reflected signals from
objects surrounding the receiver antenna.
• Multipath distorts the original signal through interference with the reflected
signals at the GPS antenna.

MULTIPATH ERROR
• There are several options to reduce the effect of multipath.
• The straightforward option is to select an observation site with no reflecting
objects in the vicinity of the receiver antenna.
• Another option to reduce the effect of multipath is to use a chock ring antenna
(a chock ring device is a ground plane that has several concentric metal hoops,
which reduce the reflected signals).

MULTIPATH ERROR
• Multipath propagation of the GPS signal is a dominant source of error in
differential positioning.
• Objects in the vicinity of a receiver antenna (notably the ground) can easily
reflect GPS signals, resulting in one or more secondary propagation paths.

MULTIPATH ERROR
• These secondary-path signals, which are superimposed on the desired direct-
path signal, always have a longer propagation time and can significantly
distort the amplitude and phase of the direct-path signal.
• Errors due to multipath cannot be reduced by the use of differential GPS,
since they depend on local reflection geometry near each receiver antenna.

MULTIPATH ERROR
• In a receiver without multipath protection, C/A-code ranging errors of 10 m
or more can be experienced.
• Multipath can not only cause large code ranging errors but also severely
degrade the ambiguity resolution process required for carrier phase ranging
such as that used in precision surveying applications.

MULTIPATH ERROR
• Multipath propagation can be divided into two classes: static and dynamic.
• For a stationary receiver, the propagation geometry changes slowly as the
satellites move across the sky, making the multipath parameters essentially
constant for perhaps several minutes.

MULTIPATH ERROR
• However, in mobile applications there can be rapid fluctuations in fractions
of a second.
• Therefore, different multipath mitigation techniques are generally employed
for these two types of multipath environments.

MULTIPATH ERROR
• In a receiver not designed expressly to handle multipath, the resulting cross-
correlation function will now have two superimposed components, one from
the direct path and one from the secondary path.
• The result is a function with a distortion depending on the relative amplitude,
delay, and phase of the secondary-path signal (in-phase secondary path or an
out-of-phase secondary path).

MULTIPATH ERROR
• The location of the peak of the function is displaced from its correct position,
resulting in a pseudo range error.
• In earlier receivers the magnitude of pseudo range error caused by multipath
can be 70–80 m.

METHODS OF MULTIPATH MITIGATION
• Multipath mitigation can be: spatial processing and time-domain processing.
• Spatial processing uses antenna design in combination with known or
partially known characteristics of signal propagation geometry to isolate the
direct-path received signal.
• Time domain processing achieves the same result by operating only on the
multipath corrupted signal within the receiver.

SPATIAL PROCESSING TECHNIQUES
Antenna Location Strategy
• A technique that minimizes ground signal reflections is to place the receiver
antenna directly at ground level.
• This causes the secondary path to have nearly the same delay as the direct
path.
• Clearly such antenna location strategies may not always be possible but can
be very effective when feasible.

Ground plane Antennas
• The most common form of spatial processing is an antenna designed to
attenuate signals reflected from the ground.
• A simple design uses a metallic ground plane disk centred at the base of the
antenna to shield the antenna from below.

• A deficiency of this design is that when the signal wavefronts arrive at the disk
edge from below, they induce surface waves on the top of the disk that then travel
to the antenna.
• The surface waves can be eliminated by replacing the ground plane with a choke
ring, which is essentially a ground plane containing a series of concentric circular
troughs one-quarter wavelength deep.

• These troughs act as transmission lines shorted at the bottom ends so that their
top ends exhibit a very high impedance at the GPS carrier frequency.
• Therefore, induced surface waves cannot form, and signals that arrive from
below the horizontal plane are significantly attenuated.

• However, the size, weight, and cost of a choke-ring antenna is significantly
greater than that of simpler designs.
• Most importantly, the choke ring cannot effectively attenuate secondary-path
signals arriving from above the horizontal, such as those reflecting from
buildings or other structures.
• Such antennas have proven to be effective when signal ground bounce is the
dominant source of multipath, particularly in GPS surveying applications.

Directive Antenna Arrays
• A more advanced form of spatial processing uses antenna arrays to form a
highly directive spatial response pattern with high gain in the direction of the
direct path signal and attenuation in directions from which secondary-path
signals arrive.

Long-Term Signal Observation
• If a GNSS signal is observed for sizable fractions of an hour to several
hours, one can take advantage of changes in multipath geometry caused by
satellite motion.
• This motion causes the relative delays between the direct and secondary
paths to change, resulting in measurable variations in the received signal.

• For example, a periodic change in signal level caused by alternate phase
reinforcement and cancellation by the reflected signals is often observable.
• It can be an effective method of multipath mitigation at a fixed site, such as
at a differential GNSS base station.

TIME-DOMAIN PROCESSING
Narrow-Correlator Technology
• Most receivers had been designed with a 2-MHz pre correlation bandwidth
that encompassed most, but not all, of the GPS spread spectrum signal
power.
• But using a significantly larger bandwidth combined with much closer
spacing of the early and late reference codes would dramatically improve the
ranging accuracy both with and without multipath.

• A 2-MHz pre correlation bandwidth causes the peak of the direct-path cross
correlation function to be severely rounded.
• An 8-MHz bandwidth has sharper peak of the direct-path cross-correlation
function is less easily shifted by the secondary-path component.

• It can also be shown that at larger bandwidths the sharper peak is more
resistant to disturbance by receiver thermal noise, even though the
precorrelation signal-to-noise ratio is increased.

Leading-Edge Techniques
• Because the direct-path signal always precedes secondary-path signals, the
leading (left-hand) portion of the correlation function is uncontaminated by
multipath.
• Therefore, if one could measure the location of just the leading part, it
appears that the direct path delay could be determined with no error due to
multipath.

MULTI PATH MITIGATION TECHNOLOGY (MMT)
• It is incorporated a number of GPS receivers manufactured by NovAtel
Corporation of Canada.
• The MMT technique reaches the theoretical performance limits for both
code and carrier phase ranging.

MULTI PATH MITIGATION TECHNOLOGY (MMT)
• It has the advantage that its performance improves as the signal
observation time is lengthened.
• MMT is based on maximum-likelihood (ML) estimation.

THE TWO-PATH ML ESTIMATOR (MLE)
• The simplest ML estimator designed for multipath is based on a two-path
model (one direct path and one secondary delayed path).
• Generalization to additional paths is straightforward, and the MMT
algorithm can be implemented for such cases.
• It is assumed that the received signal has been frequency-shifted to
baseband, and the navigation data have been stripped off.

• The two-path signal model is

• In this model the parameters,
A1 - direct path signal amplitude
φ1 - phase
τ1 - delay
A2, φ2, and τ2 are the corresponding parameters for the secondary path m(t)
is the code modulation
n(t)is the noise function is an additive zero-mean complex Gaussian noise
process with a flat power spectral density.

• It will be convenient to group the multipath parameters into the vector θ =
[A1, φ1, τ1,A2, φ2, τ2].
• Observation of the received signal r(t) is accomplished by sampling it on
the time interval [0,T ] to produce a complex observed vector r.

• The ML estimate of the multipath parameters is the vector θ of parameter
values that maximizes the likelihood function P, which is the probability
density of the received signal vector conditioned on the values of the
multipath parameters.
• In this maximization the vector r is held fixed at its observed value.

• Within the vector θ the estimates τ1 and φ1 of direct-path delay and carrier
phase are normally the only ones of interest.
• However, the ML estimate of these parameters requires that the likelihood
function P be maximized.

• A major advantage of the MMT algorithm is that its performance improves
with increasing E/N0, the ratio of signal energy E to noise power spectral
density.
• Additionally, the MMT algorithm provides ML estimates of all parameters
in the multipath model, and can utilize known bounds on the magnitudes of
the secondary paths, if available, to improve performance.

Unit III GPS Receivers and Errors

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