Global Positioning System Reference
In-Depth Information
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which implies that the variable
x
is exponentially correlated, i.e., the autocorrelation
function is decreasing exponentially (Gelb, 1974, p. 81). The symbol
τ
denotes the
correlation time, and
T
denotes the time difference between epochs
k
+
1 and
k
. The
variance of the process noise for correlation time
τ
is
2
1
e
−
2
T/
τ
q
k
q
w
k
=
E(w
k
w
k
)
=
−
(4.400)
with
q
k
being the variance of the process noise (Gelb, 1974, p. 82). The quantities
(
,q
k
) could be initially determined from data by fitting a sample mean and sample
autocorrelation function.
As
τ
0. This describes the pure white noise model with
no correlation from epoch to epoch.
x
can be thought of as a random constant that is
a nondynamic quantity.
As
τ
approaches zero, then
ϕ
=
[La
[16
τ
approaches infinity, we obtain the pure random walk. Applying l'Hospital
rule for computing the limit or using series expansion, we obtain
ϕ
=
1 and
q
w
k
=
Tq
k
. The random noises
w
k
are uncorrelated.
In general, both the dynamic model (4.389) and the observation model (4.391) are
nonlinear. The extended Kalman filter formulation (Gelb, 1974, p. 187) applies to
this general case. The reader is urged to consult that reference or other specialized
literature for additional details on Kalman filtering.
Lin
—
3
——
Lon
PgE
[16
