Global Positioning System Reference
In-Depth Information
APPENDIX A
Least Squares and Weighted Least
Squares Estimates
Christopher J. Hegarty
The MITRE Corporation
T
be a column vector containing
M
unknown parameters that
are to be estimated and
y
Let
x
=
[
xx x
M
K
]
12
T
be a set of noisy measurements that are
linearly related to
x
as described by the expression:
=
[
yy y
N
K
]
12
yHxn
=+
(A.1)
where
n
=
[
nn n
N
K
]
T
is a vector describing the errors corrupting the
N
measure-
12
ments, and
H
is an
N
×
M
matrix describing the connection between the measure-
ments and
x
.
The
maximum likelihood
estimate of
x
, denoted as
$
x
, is defined as (see, for
example, [1]):
()
$
x
=
arg max
p
y x
(A.2)
x
where
p
(
y
/
x
) is the probability density function of the measurement
y
for a fixed
value of
x
.
If the measurement errors, {
n
i
}, for
i
=
1, …,
N
, are identically Gaussian distrib-
2
uted with zero-mean and variance
, and furthermore if errors for different mea-
surements are statistically independent, then (A.2) becomes:
1
2
1
−
yHx
−
$
x
=
arg max
e
2
2
σ
(
)
N
2
(A.3)
x
2
πσ
2
=
arg min
yHx
−
x
2
The solution to (A.3) can readily be found by first differentiating
yHx
−
$
with
respect to
x
:
$
d
d
2
$
$
T
T
x
yHx HHxHy
−
=
2
−
2
(A.4)
$
663
Search WWH ::
Custom Search