Target Tracking - MATLAB Simulations for Radar Systems Design - page 443

where β m is equal to 1 τ m ⁄ . The maneuvering variance using SingerÓs model

is given by

2

A max

σ 2

-----------

= [ 14 P max + – P 0 ]

(9.139)

3

A max is the maximum target acceleration with probability P max and the term

P 0 defines the probability that the target has no acceleration.

The transition matrix that corresponds to the Singer filter is given by

1
β 2

------

1 T ( – 1 ++ β m T ρ m )

Φ =

(9.140)

1
β m

01 ------ ( 1 ρ m – )

0 0 ρ m

Note that when is small (the target has constant acceleration),
then Eq. (9.140) reduces to Eq. (9.114). Typically, the sampling interval is
much less than the maneuvering time constant ; hence, Eq. (9.140) can be
accurately replaced by its second order approximation. More precisely,

T β m = T τ m ⁄

T

τ m

T 2

1 T ⁄ 2

Φ =

(9.141)

01 T 1 ( – T ⁄ 2τ m )

00 ρ m

The covariance matrix was derived by Singer, and it is equal to

C 11 C 12 C 13

2σ 2
τ m

----------

C =

C 21 C 22 C 23

(9.142)

C 31 C 32 C 3 3

where

2β 3 T 3
3

1
2β 5

– 2β m T

4β m Te β m T

–

σ 2 2 β 2 T 2

---------

C 11 == 1 – e + 2β m T + --------------- – –

(9.143)

1
2β 4

– 2β m T – β m T

2β m Te β m T

–

β 2 T 2 C 12 = C 21 = --------- [ e + 1 – e + – 2β m T + ]

(9.144)

1
2β 3

– 2β m T

– β m Te β m T

–

---------

C 13 = C 31 = [ 1 – e ]

(9.145)