GPS Software Receivers - Fundamentals of Global Positioning System Receivers: A Software Approach - page 205

where t is obtained from Equation (9.12) and ρ i is the pseudorange to

satellite i .
A computer program ( p9 6 ) is listed at the end of this chapter to illustrate the
calculation of the satellite positions.

9.12 CALCULATION OF USER POSITION IN CARTESIAN
COORDINATE SYSTEM

The calculation of user position is discussed in Chapter 2. The inputs are the
positions of the satellites and the pseudoranges. Theoretically, the user position
can be solved from Equation (2.5) as
ρ 1 = (x 1 − x u ) 2

+ (y 1 − y u ) 2 + (z 1 − z u ) 2 + b u

(x 2 −

ρ 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 (9.20)

However, this equation is difficult to solve. A linearized version of Equation (2.7)
can be used to solve the user position through iteration as

(x i − x u )δx u + (y i − y u )δy u + (z i − z u )δz u

δρ i = (x i − + δb u

x u ) 2 + (y 1 − y u ) 2 + (z i − z u ) 2

(x i − x u )δx u + (y i − y u )δy u + (z i − z u )δz u
ρ i −

= + δb n (9.21)

b n

Following the steps in Section 2.6 and using program ( p2 1 ) in Chapter 2, the
user position x u , y u , z u can be found in the Cartesian coordinate system.

9.13 ADJUSTMENT OF COORDINATE SYSTEM OF SATELLITES

As discussed in Section 4.10, the earth-centered, earth-fixed coordinate system
is a function of time. The time used to calculate the position of a satellite and
the time used to calculate user position are different. The time used to calculate
the satellite position should be adjusted to be the same time for calculating user
position. The following three equations are used in an iterative way to perform
the adjustment.
First the pseudorange and the transit time can be found from Equation (4.48) as

(x

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

(x

t t = − x u ) 2 + (y − y u ) 2 + (z − z u ) 2 /c (9.22)