Workshop proceeding - final.pdf - Faculty of Information and ...

As illustrated in Fig.4, the structure information of the ambiguous range-Doppler maps can be well
exploited to limit the search region to a small set of admissible locations and ambiguity numbers.
Limiting the search region can reduce the computational burden.
Once the merit function of last frame has been calculated, the outer maximization of (29) is
performed to yield the last location and ambiguity number estimates
( xˆ
, mˆ
) = argmax I ( x , m )
(35)
k k k k k
xk,
mk
The previous location and ambiguity number estimates are found by tracing backwards from ( xˆ
, ˆ
k
mk)
.
A final remark is that the merit function in (34) consists of two parts, the measurement
information part and the transition probability information parts (including mode conditioned state
transition and state dependent mode transition). When the noise σ
2 is known to be very high, the
measurement information part represented by the first two terms is trivial. As a consequence, the
transition probability information parts are much more relied to determine the target trajectory. When
there is no prior information about the target dynamics, all transition probabilities from feasible
previous state and mode to the current state and mode can be assumed equal. The first two terms in the
maximization can be treated as constants and dropped.
5.2 Unknown Nuisance Parameters
k
When the nuisance parameter B is unknown, we assume it’s an unknown deterministic sequence.
k
k
In fact, the target power sequence U = ( u1,..., u k
) in B could also be considered as a Markov sequence,
but for TBD application, this information is weak comparing to the information on the Markovian
k
k
k
property of X and Μ . For the deterministic parameter B without a priori information, its MAP
k
k
estimate is equivalent to its ML estimate. The MAP estimates of X and Μ , and ML estimates of
k
k k k k
B can be obtained by maximizing the conditional joint PDF p( Z , X , Μ | B ).
ˆ k
( , ˆ k
, ˆ k k k k k
X Μ B ) = arg max p( X , Μ , Z | B ) (36)
,
k k k X Μ , B
The above maximization is split into two parts
ˆ k ˆ k ˆk k k k k
( X , Μ , B ) = argmax
⎡
max p( , , | )
⎤
, k 1
,
k 1
k ,
k
k m ⎢
X Μ Z B
x − −
⎣X Μ B
⎥
(37)
⎦
The inner maximization is denoted by
( , ) max ( k , k , k | k
Γ x m = p X M Z B ) (38)
k k k
,
X
k−1 Mk−
1 , Bk
The desired iterative relation is obtained using Bayes’ theorem to express Γ
k
k
of Γ ( x , m ). Since X is independent of B , from Bayes’ theorem it follows that
k k k
p
k+ 1 k+ 1 k+ 1 k+
1
( X , Μ , Z | B )
k+
1 k k k k+1
=( p zk+ 1, xk+ 1, mk+
1| xk, mk, B ) p( Z , X , Μ | B )
zk +1
xk + 1
b
k + 1
, while k + 1
( x , m
k+ 1 k+ 1 k+
1
) in terms
Since the depends only on and
x depends on xk
and mk
; m
k + 1
depends
on x and m , the first term on the right side of (39) is broken down into
k k
k+
1
( zk+ 1, xk+ 1, k+
1| xk, k, B )
p m m
(40)
= p( zk+ 1| xk+ 1, bk+ 1) p( xk+ 1| xk, mk) p( mk+
1| xk, mk)
Using (39) and (40), we can write Γk+ 1( x
k+ 1,
mk+
1)
in the form
Γk+ 1( xk+ 1, mk+
1)
k+ 1 k+ 1 k+ 1 k+
1
= max p( X , Μ , Z | B )
,
Xk , Μk Bk+
1
= max p( zk+ 1| xk+ 1, bk+ 1) max { p( xk+ 1| xk, mk) p( mk+
1| mk, xk) Γk( xk, mk)
}
bk+
1
xk,
mk
which is the required iterative equation. We work with the natural logarithm of (41) to transform the
merit function into an additive function
Γk+ 1( xk+ 1, mk+
1)
= max ln p( z | x , b ) +
(42)
bk
+ 1
xk,
mk
k+ 1 k+ 1 k+
1
{ p x x m + p m m x +Γ x m }
max ln ( | , ) ln ( | , ) ( , )
k+ 1 k k k+
1 k k k k k
(39)
(41)
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