Target Tracking - MATLAB Simulations for Radar Systems Design - page 431

x 0 () x 0 + () 2 x 0 – ()

ßß s () = -------------------------------------------------------

T 2

Using Eq. (9.63) the state transition matrix for the αβγ filter is

T 2
2

-----

1 T

Φ =

(9.114)

01 T
0 0 1

The covariance matrix (which is symmetric) can be computed from Eq. (9.76).
For this purpose, note that

α
β T

K =

⁄
γ T 2

(9.115)

⁄

G =

(9.116)

1 00

and

) T 2

1 α – ( 1 α – ) T ( 1 α – ⁄ 2

AI = ( – G )Φ =

(9.117)

– β T ⁄ – β + 1 ( 1 β 2 – ⁄ ) T

T 2 – 2γ ⁄ – 2γ ⁄ T ( 1 – γ )

Substituting Eq. (9.117) into (9.76) and collecting terms the VRR ratios are
computed as

2β 2α 2 ( + 2β 3αβ – ) αγ 42α – ( – – β )

( VRR ) x = ----------------------------------------------------------------------------------------------

(9.118)

( 42α – – β ) 2αβ αγ ( + – 2γ )

4β 3 4β 2 γ 2γ 2

– + ( 2 α – )

( VRR ) ß = -----------------------------------------------------------------------------

(9.119)

T 2 ( 42α – – β ) 2αβ αγ ( + – 2γ )

4βγ 2

( VRR ) ßß = -----------------------------------------------------------------------------

(9.120)

T 4

( 42α – – β ) 2αβ αγ ( + – 2γ )

As in the case of any discrete time system, this filter will be stable if and only if
all of its poles fall within the unit circle in the z-plane.

The αβγ characteristic equation is computed by setting