Probabilistic and Statistical Methods - Information Fusion in Signal and Image Processing - page 85

identically distributed sequence. Simple calculations lead to the expression of the
gradient vectors M k ( N k , respectively) of β k ( r k , respectively) with respect to X :

⎧
⎨

1
r k

k sin β k ) ∗ , M k = (cos β k , − sin β k ,k cos β k , −

[6.43]

⎩

N k = (sin β k , cos β k ,k sin β k ,k cos β k ) ∗ .

Assuming that the estimates for the bearing and the distance are independent, the
calculation of the Fisher matrix related to the estimate of the state X is a routine

exercise that leads us to:

1
σ β,k M k M k +

.

FIM =
k

δ r,k

σ r,k N k N k

[6.44]

In equation [6.44], δ r,k is equal to 1 when an active measurement is available and
otherwise to 0. We have to consider any number of active measurements, but it can
be shown [LEC 00] that we can simply consider the cases: one active measurement
out of T , two out of T and finally three out of T . We then have the following results
[LEC 00].

P ROPOSITION 6.1. Let det( FIM τ,T,β,r ) be the determinant of the matrix FIM asso-
ciated with two active measurements (separated by τ ) and T passive measurements,
then:

det FIM τ,T t ) 2 1+ β 2 − tt + τ ( t + t ) 2 . [6.45]

τ 2
rσ r σ β 4

( t −

0 ≤ t<t ≤ T

P ROPOSITION 6.2. We now consider that we have three active measurements (at 0 , τ 2
and τ 3 ) and T passive measurements, then:

det FIM τ 2 ,τ 3 ,T

τ 2 τ 3 τ 2 −

τ 3 β 2 .

T
r 2 σ β σ r

[6.46]

0 <τ 2 <τ 3 ≤ T

We get the same type of result as in the case of an active measurement [LEC 00].
In fact, we can easily see that the predominant contributions are from the terms of the
type 2 active measurements among 4 and we have the following result:

t ) 2 1+ β 2 − tt + τ ( t + t ) 2 . det( FIM ) ∝ ( t − [6.47]

0 ≤ t<t ≤ T