In a lecture on basic Kalman filter, I came across the following assumption about acceleration noise.

Acceleration Noise in Normal Kalman Filter

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. enter image description here

In a latter lecture on Unscented Kalman Filter the assumption was changed as below. enter image description here

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?


1 Answer 1



Every (extended / unscented) Kalman filter starts with a signal model. The state update equation in the second image is a very different beast from the little you show of the first model in the first image.

Without seeing the complete model, it's hard to say whether it's all correct, but if the signal models are different, then certainly the state update and output equations for the filter will be different.

For example, this paper shows two EKFs for frequency tracking that use two different signal models.

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.

  • $\begingroup$ 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 $. $\endgroup$ Sep 27, 2017 at 4:09
  • 1
    $\begingroup$ @farhanhubble : Added update. $\endgroup$
    – Peter K.
    Sep 27, 2017 at 11:09

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