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3. Dynamic model for approaching targets
As concerns the choice of target dynamic model, it’s tied to the adapted tracking coordinate system.
The most natural coordinate is the Cartesian coordinate. In a Cartesian system, the target state is
described by ( x, xyy , , ) T x y
x& y&
is the velocity vector.
Φ = & & , where ( , ) k
k
k
is the target’s position; ( , ) k
2 2
Velocity vector can be described by speed and heading pair (, v α , where v = x& + y& ,
and arctan ( x / y)
α = & & .
However, in the situation under our consideration, the measurements are the reflected power on the
ambiguous range-Doppler (i.e., range-apparent Doppler) grid. Thus, the choice of the Cartesian
coordinate requires a nonlinear transformation which results in increased complexity. It’s of interest to
directly model the target dynamics in the ambiguous range-Doppler domain. As illustrated in Fig. 2,
an approaching target travels at a velocity v and headingα . Heading is defined as the angle from the
velocity direction to the north direction. β denotes the difference between the inverse bearing and
target heading,
β = α − ( θ + π)
(1)
0
) k
Fig.2. Cartesian coordinate with radar located at the original and approaching target
The range trajectory for the approaching target is
2 2 2
rt () = r + vt −2rvtcosβ
0 0
2
dr() t 1 dr () t 2
r0 |
t=
0
t |
2 t= 0
t (2)
≈ + ⋅ + ⋅
dt 2 dt
2
1( vsin β ) 2
= r0
− ( vcos β ) t+
t
2 r0
2
Let λ = c/ Ft
be the radar wavelength; fdc
= 2vcos β / λ , the Doppler centroid; fdr
=− 2( vsin β ) /( λr0
) ,
the Doppler rate. (2) can be rewritten as
λ λ 2
rt () = r0
− fdct− fdrt
(3)
2 4
The Doppler trajectory can be written as
2 dr( t)
fd()
t =− = f dc
+ f dr
t (4)
λ dt
The discrete-time target dynamic model can be obtained by sampling the above continuous-time
model. Sampling the above continuous-time models at the interval ofT R
, we get
fdk , + 1
= fd()| t
t= ( k+
1) T
= f ( 1)
R dc+ fdr k+
TR
(5)
= f + f T
dk , dr R
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