Graphics Programs Reference
In-Depth Information
IA
z
1
1
2
=
12α
(
β
)
z
+
(
1 α
)
z
=
0
(9.96)
Solving Eq. (9.96) for
z
yields
1
αβ
+
2
1
--- αβ
)
2
z
12
=
-------------
±
(
4β
(9.97)
,
and in order to guarantee stability
z
12
<
1
(9.98)
,
Two cases are analyzed. First,
z
12
are real. In this case (the details are left as
,
an exercise),
β 0
>
;
α
>
β
(9.99)
The second case is when the roots are complex; in this case we find
α 0
>
(9.100)
The system transfer functions can be derived by using Eqs. (9.79), (9.89),
and (9.90),
α
zz
αβ
(
)
-----------------
h
x
()
h
ß
()
1
α
----------------------------------------------------------------
=
(9.101)
z
2
z
2 α
(
β
)
+
(
1 α
)
β
zz
1
(
)
----------------------
T
Up to this point all relevant relations concerning the filter were made
with no regard to how to choose the gain coefficients ( and ). Before con-
sidering the methodology of selecting these coefficients, consider the main
objective behind using this filter. The twofold purpose of the
αβ
αβ
αβ
tracker can
be described as follows:
The tracker must reduce the measurement noise as much as possible.
1.
The filter must be able to track maneuvering targets, with as little residual
(tracking error) as possible.
2.
The reduction of measurement noise is normally determined by the VRR
ratios. However, the maneuverability performance of the filter depends heavily
on the choice of the parameters
αβ
and
.
A special variation of the filter was developed by Benedict and Bord-
ner
1
, and is often referred to as the Benedict-Bordner filter. The main advan-
αβ
1. Benedict, T. R. and Bordner, G. W., Synthesis of an Optimal Set of Radar Track-
While-Scan Smoothing Equations,
IRE Transaction on Automatic Control, AC-7,
July 1962, pp. 27-32.
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