Basic GPS Concept - Fundamentals of Global Positioning System Receivers: A Software Approach - page 12

where b u is the user clock bias error expressed in distance, which is related to the
quantity b ut by b u cb ut . In Equation (2.5), four equations are needed to solve

for four unknowns x u , y u , z u ,and b u . Thus, in a GPS receiver, a minimum of
four satellites is required to solve for the user position. The actual measurement
of the pseudorange will be discussed in Chapter 9.

=

2.6 SOLUTION OF USER POSITION FROM PSEUDORANGES

One common way to solve Equation (2.5) is to linearize them. The above
equations can be written in a simplified form as

(x i −

ρ i = x u ) 2 + (y i − y u ) 2 + (z i − z u ) 2 + b u ( 2 . 6 )

where i = 1, 2, 3, and 4, and x u , y u , z u ,and b u are the unknowns. The
pseudorange ρ i and the positions of the satellites x i , y i , z i are known.
Differentiate this equation, and the result is

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

δρ i = (x i − x u ) 2 + δb u

+ (y i − 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

= +

δb u (2.7)

ρ i − b u

In this equation, δx u , δy u , δz u ,and δb u can be considered as the only unknowns.
The quantities x u , y u , z u ,and b u are treated as known values because one can
assume some initial values for these quantities. From these initial values a new
set of δx u , δy u , δz u ,and δb u can be calculated. These values are used to modify
the original x u , y u , z u ,and b u to find another new set of solutions. This new set
of x u , y u , z u ,and b u can be considered again as known quantities. This process
continues until the absolute values of δx u , δy u , δz u ,and δb u are very small and
within a certain predetermined limit. The final values of x u , y u , z u ,and b u are
the desired solution. This method is often referred to as the iteration method.
With δx u , δy u , δz u ,and δb u as unknowns, the above equation becomes a set of
linear equations. This procedure is often referred to as linearization. The above
equation can be written in matrix form as

δρ 1
δρ 2
δρ 3
δρ 4

α 11 α 12 α 13 1 δx u

δy u
δz u
δb u

=

α 21 α 22 α 23 1

( 2 . 8 )

α 31 α 32 α 33 1

α 41 α 42 α 43 1

where

x i − x u
ρ i −

y i − y u
ρ i −

z i − z u
ρ i −

α i 1 = α i 2 = α i 3 = ( 2 . 9 )

b u b u b u