In Kalman filters why is it necessary to transform the systems dynamics matrix to the state transition matrix? Previously when I have implemented Kalman filters I have used the transformation 

$$
\mathbf{A(t)} = \mathcal{L}^{-1} \left( s \mathbf{I} - \mathbf{F} \right) ^{-1}
$$

to calculate the state transition matrix $\bf{A}$ from the system dynamics matrix $\bf{F}$.

Where the equation describing the state of the system is

$$
\vec{x_t} = \mathbf{F_t} \vec{x_{t-1}} + \mathbf{B_t} \vec{u_t} + \vec{w_t}
$$

and my prediction step calculation is 

$$
\vec{x_{k|k-1}} = \mathbf{A} \vec{x_{k-1|k-1}} + B\vec{u_k}
$$

For example for simple sinusoidal Kalman filter started from the following equation of motion:

$ \ddot{x} = -\omega^2x $

So my matrix equation of motion was:

$$
\begin{bmatrix}
  \dot{x} \\
  \ddot{x} \\
\end{bmatrix}
=
\begin{bmatrix}
  0 &amp; 1 \\
  -\omega^2 &amp; 0 \\
\end{bmatrix}
.
\begin{bmatrix}
  x \\
  \dot{x} \\
\end{bmatrix}
$$

and my system dynamics matrix was

$$
F = \begin{bmatrix}
  0 &amp; 1 \\
  -\omega^2 &amp; 0 \\
\end{bmatrix}
$$

I then performed the transformation $
\mathbf{A(t)} = \mathcal{L}^{-1} \left( s \mathbf{I} - \mathbf{F} \right) ^{-1}
$ to get the state transition matrix like so:

$$
A(t) = \begin{bmatrix}
  \cos(\omega t) &amp; \dfrac{\sin(\omega t)}{\omega} \\
  -\omega \sin(\omega t) &amp; \cos(\omega t) \\
\end{bmatrix}
$$

Where I then replaced $t$ with my sample time $T_s$ where $T_s = \dfrac{1}{F_s}$ where $F_s$ was my sample frequency.

I then used this to extract my sinusoidal signal.

My question is did I need to perform this transformation? And if I did, why did I have to do so?

I was reading the paper linked here where they attempt the explain the Kalman filter in simple way and they did no such transformation.
As I said in the comments, there should be no need at all to do such a transformation if all of your equations start in discrete-time.

The equation
$$
 (s \mathbf{I} - \mathbf{F} )^{-1}
$$
is just the Laplace transform of the solution for the state equation:
$$
\dot{\vec{x_t}} = \mathbf{F} \vec{x_t} + \vec{w_t}
$$

However, I really can't see how $\mathbf{F}$ works in:
$$
\vec{x_t} = \mathbf{F_t} \vec{x_{t-1}} + \mathbf{B_t} \vec{u_t} + \vec{w_t}
$$
because this is a discrete-time equation, and the equation you're looking at generating the sinewave is continuous-time.

Can you clarify?