Basic GPS Concept - Fundamentals of Global Positioning System Receivers: A Software Approach - page 11

satellites must be measured simultaneously at a certain time instance. Each satel-
lite transmits a signal with a time reference associated with it. By measuring the
time of the signal traveling from the satellite to the user the distance between
the user and the satellite can be found. The distance measurement is discussed
in the next section.

2.5 MEASUREMENT OF PSEUDORANGE ( 2 )

Every satellite sends a signal at a certain time t si . The receiver will receive the
signal at a later time t u . The distance between the user and the satellite i is

ρ iT = c(t u − t si ) ( 2 . 2 )

where c is the speed of light, ρ iT is often referred to as the true value of pseu-
dorange from user to satellite i , t si is referred to as the true time of transmission
from satellite i , t u is the true time of reception.
From a practical point of view it is difficult, if not impossible, to obtain the
correct time from the satellite or the user. The actual satellite clock time t si and
actual user clock time t u are related to the true time as

t si =

t si +

b i

t u = t u + b ut (2.3)

where b i is the satellite clock error, b ut is the user clock bias error. Besides
the clock error, there are other factors affecting the pseudorange measurement.
The measured pseudorange ρ i can be written as ( 2 )

ρ i = ρ iT + D i − c(b i − b ut ) + c(T i + I i + v i + v i ) ( 2 . 4 )

where D i is the satellite position error effect on range, T i is the tropospheric
delay error, I i is the ionospheric delay error, v i is the receiver measurement
noise error, v i is the relativistic time correction.
Some of these errors can be corrected; for example, the tropospheric delay
can be modeled and the ionospheric error can be corrected in a two-frequency
receiver. The errors will cause inaccuracy of the user position. However, the
user clock error cannot be corrected through received information. Thus, it will
remain as an unknown. As a result, Equation (2.1) must be modified as
ρ 1 = (x 1 − x u ) 2 + (y 1 − y u ) 2 + (z 1 − z u ) 2 + b u

ρ 2 = (x 2 − x u ) 2 + (y 2 − y u ) 2 + (z 2 − z u ) 2 + b u

(x 3 −

ρ 3 = x u ) 2 + (y 3 − y u ) 2 + (z 3 − z u ) 2 + b u

(x 4 −

ρ 4 = x u ) 2 + (y 4 − y u ) 2 + (z 4 − z u ) 2 + b u (2.5)