Military Communications and Information Technology: A Trusted ...

Chapter 3: Information Technology for Interoperability and Decision...
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tion. In addition, ASDs fully describe the correlations between the state estimates
at different instants of time.
All information on the target states accumulated over a time window
tk, tk , , tn
of length kn
1,
xkn ( xk, , xn)
(10)
that can be extracted from the time series of accumulated sensor data up to
k
and including time is contained in a joint density function p( x
kn| Z ), which
is called a ASD.
Iterative ASD
ASD posterior
z z z z z z z z z
0
time t
Figure 2. Schematic illustration of an iterative ASD filter and the ASD posterior. The iterative filtering
scheme includes the calculation of a prior ASD density using a fixed window size. It is based
on an ASD posterior, which yields the joint density conditioned on the entire set of measurements
Given the full posterior ASD [2], one can calculate the exact cross-covariance
to timely delayed measurements. To this end, let us consider a measurement
produced at time with t n
m t k
i.e. possibly before the `present’ time
where the time series is available and has been exploited. We would like to
understand the impact this new, but late sensor information has on the present
and the past target states xl , l n, , k,
i.e. on the accumulated state x
kn
Let
be a measurement of the observed object state at time characterized by
a Gaussian likelihood function, which is defined by a measurement matrix and
a corresponding measurement error covariance matrix We further renumber
the target states xk, , xn
such that xk, , xm, , xn : xkmn
: :
are consistent with
their time stamps ( tl) lk, , m, ,
n.
By an application of continuous time retrodiction (see [3], e.g., for a detailed
discussion), it is well possible to extend the posterior density of the state x
kn
to
a prior density of the extended state x
kmn : :
[4]. One obtains a single Gaussian density
with a joint expectation vector including the estimates for all single states:
k
px ( | Z) N ( x ; x , P ),
(11)
kmn : : kmn : : kmnk : : | kmnk : : |
where the expectation vector accumulates the target estimates for each state within
the time window:
x ( x , , x , , x ).
(12)
k: mnk : | kk | mk | nk |