Military Communications and Information Technology: A Trusted ...

284 Military Communications and Information Technology...
time with respect to the minimum mean squared error (MMSE) yields the Kalman
filter [1] The Kalman filter consists of the two steps prediction:
x F x
(3)
,
kk | 1 kk | 1 k1| k1 and filtering:
P F P F Q
kk | 1 kk | 1 k1| k1 kk | 1 kk | 1
(4)
x x W
(5)
kk | kk | 1 kk | 1 k,
(6)
,
k
zk Hx
k kk | 1
W P H S
kk | 1
kk | 1 k k
,
(7)
S HP H R
(8)
k
k kk | 1 k
k,
P P W SW
.
kk |
kk | 1
kk | 1 k kk | 1 (9)
However, in network constrained scenarios Kalman filter assumptions are
often not satisfied. In particular, imperfect communication leads to:
a) measurements, which arrive in a timely disordered way,
b) insufficient bandwidth such that not all measurements can be transmitted,
and
c) synchronization and sensor registration errors.
As a consequence, there is a need for sophisticated algorithms which are
suited to multi-sensor scenarios and robust against significant communication
constraints. In this paper, we present two state-of-the-art Kalman filters for tracking
object parameters in such scenarios.
III. Kalman filter processing for delayed measurements
In most target tracking algorithms, the characteristics of conditional probability
k
densities p( xl
| Z ) of target states are calculated, which describe the available
knowledge of the target properties at a certain instant of time given a time
series of imperfect sensor data accumulated up to time In certain applications,
however, the kinematic target states xk
, , xn,
n k,
accumulated over a certain time
window from a past instant of time up to the present time is of interest.
The statistical properties of the accumulated state vectors are completely described
by the joint probability density function conditioned on the measurement data,
k
p( xk, , xn
| Z ). These densities are called Accumulated State Densities (ASDs).
By marginalizing them, the standard filtering and retrodiction densities directly
result; in other words, ASDs provide a unified description of filtering and retrodic-