Fundamentals of Satellite Navigation - Understanding GPS: Principles and Applications - page 58

where meas is the vector containing the pseudorange measurement errors and

x is the

vector representing errors in the user position and receiver clock offset.
The error contribution x can be minimized by making measurements to more

than four satellites, which will result in an overdetermined solution set of equations
similar to (2.33). Each of these redundant measurements will generally contain inde-
pendent error contributions. Redundant measurements can be processed by least
squares estimation techniques that obtain improved estimates of the unknowns.
Various versions of this technique exist and are usually employed in today's receiv-
ers, which generally employ more than four user-to-satellite measurements to com-
pute user PVT. Appendix A provides an introduction to least squares techniques.

2.5 Obtaining User Velocity

GPS provides the capability for determining three-dimensional user velocity, which
is denoted u . Several methods can be used to determine user velocity. In some receiv-

ers, velocity is estimated by forming an approximate derivative of the user position,
as shown here:

&

() ()

u t −

−

u t

d
dt

u

2 1

& u ==

t t

2 1

This approach can be satisfactory provided the user's velocity is nearly constant
over the selected time interval (i.e., not subjected to acceleration or jerk) and that the
errors in the positions u ( t 2 ) and u ( t 1 ) are small relative to difference u ( t 2 ) u ( t 1 ).

In many modern GPS receivers, velocity measurements are made by processing
carrier-phase measurements, which enable precise estimation of the Doppler fre-
quency of the received satellite signals. The Doppler shift is produced by the relative
motion of the satellite with respect to the user. The satellite velocity vector v is com-
puted using ephemeris information and an orbital model that resides within the
receiver. Figure 2.18 is a curve of received Doppler frequency as a function of time
measured by a user at rest on the surface of the Earth from a GPS satellite. The
received frequency increases as the satellite approaches the receiver and decreases as
it recedes from the user. The reversal in the curve represents the time when the
Doppler shift is zero and occurs when the satellite is at its closest position relative to
the user. At this point, the radial component of the velocity of the satellite relative to
the user is zero. As the satellite passes through this point, the sign of

−

f changes. At
the receiver antenna, the received frequency, f R , can be approximated by the classical
Doppler equation as follows:

∆

( )

1 va

⋅

r

f = f − (2.36)

R T

c

where f T is the transmitted satellite signal frequency, v r is the satellite-to-user relative
velocity vector, a is the unit vector pointing along the line of sight from the user to
the satellite, and c is the speed of propagation. The dot product v r · a represents the