# Kalman filter - understanding the noise covariance matrix

What is the significance of the noise covariance matrices in the Kalman Filter framework?

I am referring to:

• process noise covariance matrix Q, and
• measurement noise covariance matrix R

at any time step t.

How do I interpret these matrices? What do they represent? Do they talk about how one observation's noise varies with respect to another observation's noise in the state vector?

• Excellent intuitive explanation! I also have two questions 1. First what is the meaning of covariance of let's say 1,3 element of the acceleration covariance matrix? 2. Secondly, how does one tune the observation noise covriance matrix for the firs step of the algortihm? If that requires high computational effort or mathematics, what are some good typical values when trying to observe a multi degree of freedom vibrating system ? Thank you very much. – george p Nov 24 '17 at 16:30
• @georgep Please NEVER post follow-up questions as an answer. Please ask a new question, but perhaps link to this question when you do. – Peter K. Nov 24 '17 at 21:42

Q is in state space, and R is in measurement space. In the example above, our state might be position only $[x, y]^T$, and measurement space is velocity $[v]$. That is problematic because that is not velocity in terms of x and y - you need the heading to convert. The Kalman filter matrix H is used to do that conversion, and in nonlinear systems you tend to have to linearize that in some manner.