# Is acceleration noise modelled differently in EKF and UKF Kalman Filters?

In a lecture on basic Kalman filter, I came across the following assumption about acceleration noise. Every component of the noise vector $\nu$ is itself a product of time and acceleration values. Nothing is known about the accelerations $a_x$ and $a_y$ but each component of $\nu$ is assumed to have zero mean and every pair of components $\nu_j$, $\nu_k$ has some known variance $\sigma^2_{\nu_j\nu_k}$. As a consequence the last term in each of the four equations is set to zero (not shown here). The covariance matrix Q contains time though as in the screenshot below. In a latter lecture on Unscented Kalman Filter the assumption was changed as below. Here the components of noise vector $\nu$ do not contain any effect of time. These time-free components are assumed to have mean zero and covariance $Q$. As a result their effect on the mean state vector $x$ cannot be set to zero. Also $Q$ is independent of time.

Is it standard practice to make different kind of assumptions?

Regarding incorporation of the $\Delta t$ terms in the acceleration, that does seem a bit odd. Usually those terms form part of either the state transition matrix or the input matrix. See this answer for one way to do that for a simpler model.
• I do understand that every model will have a different set of assumptions, what I specifically want to know is the rationale behind assuming the acceleration * time product to be Gaussian in case of EKF and acceleration alone to be Gaussian in the UKF case. Since acceleration is assumed to be produced by a combination of random causes it makes sense to assume that $a_x$ or $a_y$ is Gaussian. I am able to follow this reasoning and the resulting UKF equations. Similar to this, how do we justify the Gaussian assumption for $a_x * \Delta t$ and $a_x * {\Delta t}^2$. – farhanhubble Sep 27 '17 at 4:09