Graphics Programs Reference
In-Depth Information
α
β
T
K
=
⁄
γ
T
2
(9.69)
⁄
One of the main objectives of a tracking filter is to decrease the effect of the
noise observation on the measurement. For this purpose the noise covariance
matrix is calculated. More precisely, the noise covariance matrix is
)
x
t
C
nn
(
)
=
E
x
nn
{
(
(
)
(
nn
)
}
;
y
n
=
v
n
(9.70)
where indicates the expected value operator. Noise is assumed to be a zero
mean random process with variance equal to
E
σ
2
. Additionally, noise measure-
ments are also assumed to be uncorrelated,
δσ
2
nm
=
Ev
n
v
m
{
}
=
(9.71)
0
nm
≠
Eq. (9.65) can be written as
x
nn
(
)
=
Ax
n
(
1
n
1
)
+
K
y
n
(9.72)
where
AI
=
(
G
)Φ
(9.73)
Substituting Eqs. (9.72) and (9.73) into Eq. (9.70) yields
)
t
C
nn
(
)
=
E
Ax
n
1
{
(
(
n
1
)
+
K
y
n
)
Ax
n
(
(
1
n
1
)
+
K
y
n
}
(9.74)
Expanding the right hand side of Eq. (9.74) and using Eq. (9.71) give
)
A
t
K
σ
2
K
t
C
nn
(
)
=
AC
n
(
1
n
1
+
(9.75)
Under the steady state condition, Eq. (9.75) collapses to
ACA
t
K
σ
2
K
t
C
nn
(
)
=
+
(9.76)
where
C
is the steady state noise covariance matrix. In the steady state,
C
nn
(
)
=
C
n
1
(
n
1
)
=
C
for any n
(9.77)
Several criteria can be used to establish the performance of the fixed-gain
tracking filter. The most commonly used technique is to compute the Variance
Reduction Ratio (VRR). The VRR is defined only when the input to the tracker
is noise measurements. It follows that in the steady state case, the VRR is the
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