Data Processing for Positioning - A Software-Defined GPS and Galileo Receiver: A Single-Frequency Approach - page 120

2 T

the usual way, so that is the sum of squares of the m

separate errors, minimizing this quadratic gives the normal equations
A T A

e = ( b − A x ) ( b − A x )

A T b A T A ) − 1 A T b x ˆ = or x ˆ = ( (8.23)

and the error vector is

ˆ = − ˆ .

e b A x (8.24)

The covariance matrix for the parameters x is ˆ

e T ˆ e ˆ

2

A T A ) − 1

2
0

x = σ

0 (

with σ =

n .

(8.25)

m −

The linearized observation equation (8.21) can be rewritten in a vector formula-
tion

⎡ ⎤

X i

1

⎣ ⎦ −

Y i

X k Y k Z k − X i , 0 − Y i , 0 − Z i , 0

P i

k
i

cdt k T i I i + e i .

− − −

= ρ

0 +

+ +

,

k
i , 0

k
i , 0

k
i , 0

ρ ρ ρ

Z i
cdt i

(8.26)
We rearrange this to resemble the usual formulation of a least-squares problem
A x = b :

⎡
⎣

⎤
⎦ =

X i

1

Y i

X k Y k Z k − X i , 0 − Y i , 0 − Z i , 0

P i

k
i

cdt k T i I i − e i .

− − −

− ρ

0 +

− −

i i i

,

ρ ρ ρ

Z i
cdt i

, 0 , 0 , 0

(8.27)

A unique solution cannot be found from a single equation. Let b i P i

k
i ,

= − ρ

0 +

cdt k T i I i e i . Then the final solution comes from − − −

⎡
⎣

⎤
⎦

X 1 Y 1 Z 1 − X i , 0 − Y i , 0 − Z i , 0

− − − 1

1
i , 0

1
i , 0

1
i , 0

ρ ρ ρ

X 2 Y 2 Z 2

− X i , 0 − Y i , 0 − Z i , 0

⎡
⎣

⎤
⎦ =

− − − 1

X i , 1

2
i , 0

2
i , 0

2
i , 0

ρ ρ ρ

Y i , 1

X 3 Y 3 Z 3

=

− X i , 0 − Y i , 0 − Z i , 0

− .

A x b e (8.28)

− − −

1

Z i , 1
cdt i , 1

3
i , 0

3
i , 0

3
i , 0

ρ ρ ρ

. . . .

X m Y m Z m − X i , 0 − Y i , 0 − Z i , 0

− − − 1

m
i ,

m
i ,

k
i , 0

ρ ρ

ρ

0 0

If m Z i , 1 . This has to be added to

the approximate receiver position to get the next approximate position:
X i , 1 =

≥ 4, there is a unique solution: X i , 1 , Y i , 1 ,

X i , 0 + X i , 1 ,

Y i , 1 = Y i , 0 + Y i , 1 , (8.29)

Z i , 1 = Z i , 0 + Z i , 1 .