# Gaussian Noise Covariance Matrix in the Extended Kalman Filter

In Simultaneous Localization and Mapping: Part I, the Extended Kalman Filter is described on page 5. I'm confused about where it says "$w_k$ are additive, zero mean uncorrelated Gaussian motion disturbances with covariance $Q_k$".

To state an assumption: I think that $w_k$ is a single motion disturbance at time k, although I'm not positive about this.

My question is, given that covariance is the measure of how much two random variables change together, what are the two random variables for $Q_k$?

For the Kalman Filter, $w_{k}$ acts like the noise that corrupts your observations. In a perfect world, you could observe the motion without any problems, but in reality, the sensor or measuring device isn't perfect so you get random extra jitters or artifacts that are just random error. Since you're estimating many different parameters of a model, though, some of the errors might affect two or more different variables. That's why you have to consider the covariance matrix, $Q_{k}$. It tells you what that random-noise-structure looks like for your given problem.
I had thought the covariance $Q_k$ was a scalar, not a matrix describing the relationship between all the parameters of the model. –  munkhd Mar 15 '12 at 18:59