Global Positioning System Reference
In-Depth Information
The pseudorange errors are considered to be random variables, and (7.36)
expresses
d
x
as a random variable functionally related to
d
. The error vector
d
is
usually assumed to have components that are jointly Gaussian and to be zero mean.
With the geometry considered fixed, it follows that
d
x
is also Gaussian and zero
mean. The covariance of
d
x
is obtained by forming the product
d
x
d
x
T
and comput-
ing an expected value. By definition, one obtains
()
[
]
T
cov
d
x
=
Ed d
x x
(7.37)
E
[
d
x
d
x
T
] denotes the covariance of
d
x
and
E
represents the expec-
tation operator. Substituting from (7.36) and viewing the geometry as fixed, one
obtains
where cov(d
x
)
=
[
]
(
)
(
)
−
1
−
1
()
cov
d
x
=
Ed k
K
TT
K HHHHHH
=
E
T
T
d d
T
T
(7.38)
(
)
()
(
)
−
1
HHH
−1
T
T
T
=
HH H
cov
d
Note that in this computation, (
H
T
H
)
−1
is symmetric. [This follows from an
application of the general matrix relations (
AB
)
T
(
A
T
)
−1
, which
are valid whenever the indicated operations are defined.] The usual assumption is
that the components of
d
are identically distributed and independent and have a
variance equal to the square of the satellite UERE. With these assumptions, the
covariance of
d
=
B
T
A
T
and (
A
−1
)
T
=
is a scalar multiple of the identity
()
2
cov
d
=
×
σ
(7.39)
nn UERE
where
I
n
×
n
is the
n
×
n
identity matrix. Substitution into (7.38) yields
()
(
)
−
1
T
2
cov
d
x HH
=
σ
(7.40)
UERE
Under the stated assumptions, the covariance of the errors in the computed position
and time bias is just a scalar multiple of the matrix (
1
. The vector
d
x
has four
components, which represent the error in the computed value for the vector
x
T
=
T
−
HH
)
(
x
u
,
y
u
,
z
u
,
ct
b
). The covariance of
d
x
is a 4
×
4 matrix and has an expanded representation
σ
2
σ
2
σ
2
σ
2
x
x
y
x
z
x
ct
u
u
u
u
u
u
b
σ
2
σ
2
σ
2
σ
2
()
x
y
y
y
z
y
ct
cov
d
x
=
(7.41)
u
u
u
u
u
u
b
σ
2
σ
2
σ
2
σ
2
xz
y z
z
z ct
uu
uu
u
u b
2
2
2
2
σ
σ
σ
σ
xct
y ct
z ct
ct
u
b
u
b
u
b
b
The components of the matrix (
H
T
H
)
−1
quantify how pseudorange errors translate
into components of the covariance of
d
x
.
Dilution of precision parameters in GPS are defined in terms of the ratio of com-
binations of the components of cov(
d
x
) and
σ
UERE
. [It is implicitly assumed in the
DOP definitions that the user/satellite geometry is considered fixed. It is also
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