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

⎧π0
if j = i and Cx
ij k
> δij
⎪
π
ij
( xk ) = ⎨1 − π0
if j = i−1 and Cijx k
≤δij
(14)
⎪
⎩0 else
whereπ 0
is the probability of staying in the same mode, the guard thresholdδ ij
is determined by the
level of Doppler rate and its incremental noise rate increment.
Due to the uncertainties in target dynamics, δ
ij
is well modeled as a random variable. For
simplicity, we assume δ ij
is a Gaussian distributed variable with mean δ ij
and variance Σ ij
, denoted
byδ N ( δ ; δ , Σ ), then the conditional mode transition probability from mode i to mode j = i − 1 is
ij ij ij ij
given by
π ( x ) = p δ ≥ C x | m = i,
x
( )
ij k ij ij k k k
(15)
Sinceδ ij
is Gaussian distributed, then (15) is given by
( δ )
π ( x ) = 1 −Φ C x − ; Σ (16)
ij k ij k ij ij
where Φ( δ ij
; Σ ij ) denotes a Gaussian cumulative probability function (CDF) with mean δ ij
and
variance
Σ ij
.The probability that target stays the same ambiguity number is given by
π ( x ) = 1 −π
( x )
(17)
ii k ij k
Fig.3. Illustration of state-dependent ambiguity number transition
4. Measurement Statistical Model
Let x()
t be the signal transmitted by the radar. For simplicity, the transmitted signal is a
rectangular pulse train x() t = a()exp( t j2 π Ft t ), where F t
is the carrier frequency and at () is the pulse
envelope of the form
⎧⎪ 1:0 ≤mod( t,
T p ) ≤τ
(18)
at () = ⎨
⎪⎩ 0:others
whereτ is the pulse width.
The received baseband signal during the kth
visit, k = 1,..., K the can be written as
H : y () t = g () t
0
k
k
Nd
⎛
2r
k ⎞
H1
: yk() t = ∑Aa k ⎜t−nTp−( k−1) TR− exp ( j2π
f t
, ) + g ()
dk k
t
n=
0
c
⎟
⎝
⎠
(19)
14