# Why discretize a continuous transition matrix in Kalman Filter?

In the Kalman filter toolbox at http://becs.aalto.fi/en/research/bayes/ekfukf/cwpa_demo.html the example code shows that a function lti_disc is called, which is essentially a matrix exponential function. What is the purpose of such matrix exponentiation? Given a really simple transition model such as, e.g.

F = [ 1 0 ;
0 1 ] ;


and doing the matrix exponentiation with lti_disc will always give a different transition matrix A, depending on the values of the Q process noise covariance, which is also an input to the lti_disc function.

• not quite sure what integration has to do with discretization? – Marcus Müller Mar 17 at 9:38

1. The resulting $$\bf{A}$$ matrix does not depend on the process noise
2. The $$\bf{A}$$ matrix only depends on the size of the time step.
3. The $$\bf{Q}$$ matrix only depends on the size of the time step, and the value of the continuous noise covariance, ie. diag(q) in the example
You can use discrete noise kinematic models, they are probably a bit easier to understand. The continuous noise model assume that the noise is applied throughout the time step interval i.e. from time $$t_0$$ to $$t_1$$. The discrete model assumes the noise is applied instantaneously at times $$t_0$$ and $$t_1$$ (and so on).