Military Communications and Information Technology: A Trusted ...

Chapter 3: Information Technology for Interoperability and Decision...
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in the fusion center. In particular, it is possible to modify all local tracks such that
they become decorrelated. The cross-correlations occur because of a common evolution
covariance in every prediction step [5]. This is due to the fact that all sensors
sites observe the same target. In [6] it is shown that a central Kalman filter can be
calculated in a distributed manner by decorrelating the local tracks. This is achieved
by calculating the global estimation error covariance of the system. The proposed
distributed Kalman-type processing scheme makes essentially use of the fact that
the sensor measurements do not enter into the update equation for the estimation
error covariance matrices. In particular, the covariance matrices of all sensors
can be calculated at each individual sensor site without any further communication
(provided the relevant parameters of all sensors are known at each sensor site).
For the sake of notational simplicity, let all S sensors be equally aligned and
synchronized with the same data update rate. However, these assumptions are not
essential and can well be relaxed. Furthermore, we assume the measurement error
covariance matrices { R
s }
S
k s
of all individual sensors to be known to each local sensor
processor. As mentioned above, the proposed distributed processing scheme aims at
establishing decorrelated local tracks, such that fusing them yields exactly the result
of central Kalman filter processing. Let us assume, a set of decorrelated local tracks
at time are given which have processed all sensor data up to time Then, as all
densities involved are Gaussians, we can write the fused track as the following product:
S
k s s
l lk |
N (
l; xlk |
,
lk |
s1
px ( | Z) c x P)
,.
(21)
In the sequel, as well as in most applications, it is unnecessary to calculate
the normalization constant c
lk
explicitly. By virtue of the factorization lemma for
Gaussians, this product representation can be transformed into a single Gaussian:
S
k s s
l lk |
N
l; lk |
,
lk |
s1
px ( | Z) c ( x x P)
(22)
N ( x ; x , P ),
(23)
l lk | lk |
with an expectation vector x
lk
and a covariance matrix P
lk
obtained by fusing x
lk
and P s , s1, , S,
according to following equations:
lk
S
1 s 1
lk |
Plk
|
s1
P
(24)
S
s
s
lk | lk | lk | lk |
s1
x P ( P x ).
(25)
Convex combinations of this type are fundamental in almost all data fusion
applications (see e.g. [7]). As previously stated, under conditions where Kalman