Target Tracking - MATLAB Simulations for Radar Systems Design - page 420

where defines the discrete time and is the sampling interval. All other
vectors and matrices were defined earlier. The homogeneous solution to the
system defined in Eq. (9.38), with initial condition

n nT T

x n () , is

x () A nn 0

–

= x n ()

(9.40)

In this case the state transition matrix is an nn × matrix given by

A nn 0

–

Φ nn 0 ( , ) Φ nn 0 = ( – ) =

(9.41)

The following is the list of properties associated with the discrete transition
matrix

Φ n ( + 1 – n 0 ) = A Φ nn 0 ( – )

(9.42)

Φ n 0 ( – n 0 ) () I = =

(9.43)

Φ n 0 ( + 1 – n 0 ) = Φ () A =

(9.44)

Φ n 2 ( – n 0 ) n 2 = ( – n 1 ) Φ n 1 ( – n 0 )

(9.45)

) 1

–

Φ n 0 ( – n 1 = ( n 1 – n 0 )

(9.46)

) n () Φ 1

–

Φ n 1 ( – n 0 = n () (9.47)

The solution to the general case (i.e., non-homogeneous system) is given by

n –

∑

1

x () Φ nn 0 = ( – ) x n () Φ nm 1 + ( – – ) Bw ()

(9.48)

m 0 =

It follows that the output is given by

n –

∑

1

y () C Φ nn 0 = ( – ) x n () C Φ nm 1 + ( – – ) Bw () Dw () +

(9.49)

m 0 =

where the system impulse response is given by

n –

∑

1

h () = C Φ nm 1 ( – – ) B δ () D δ () + (9.50)

m 0 =

Taking the Z-transform for Eqs. (9.38) and (9.39) yields