Extensive writeup:

http://home.earthlink.net/~david.schultz/rnd/2004/KalmanApogeeII.pdf

I took a lot of math in college but not enough to grasp all of this.

I think the only reason why most people don't grasp it - is because there appears to be never one source of information that 'fully' guides everybody from beginning to end.....with step-by-step examples of collecting various measurement data, collecting background noise data, then calculating covariance matrices from the data sets, then choosing starting values (starting estimates), and using the formulae and the obtained matrices/data to compute things like a projected/future values (such as 0.001 second ahead) using a 'kalman gain matrix'. A lot of sources explain the various matrix equations in wishy-washy ways as well. Can't blame people for not understanding it. There are at least a few internet sources, where the authors do provide enough detail to get people into the subject. But usually, it's either necessary to gather a whole bunch of sources together and figure it all out (including all the statistic mathematical manipulations/derivations) yourself, or ask somebody that already understands the area very well (to iron out uncertainties or questions).

The word 'fusion' used in kalman discussions is kind of wishy-washy in itself. But I guess they have to give a word for it heheh. I prefer to just call it information combining, or estimation based on gathering of information (with statistical considerations).

Also..... one more thing..... the equation at the top of page 14 at this link here ...

http://home.earthlink.net/~david.schultz/rnd/2004/KalmanApogeeII.pdfhas three vertical lines in the expression, plus three dots.

I think it means ... n = the number of states = 3, and the vertical lines are just separators for sub-matrices in-between, to be stitched together. And the pattern is:

{(Phi_T)^0}H_T {(Phi_T)^1}H_T {(Phi_T)^2}H_T

We stop at (Phi_T)^2 because that term represents (Phi_T)^(n-1) where n = 3

The "..." just means : keep building up the power of Phi_T until we get to a power index of n-1.

Update: I know what's going on now. The matrix is meant to be the 'observability matrix', so it involves the output matrix C... or in this case 'H', and also involves the state transition matrix A... in this case 'Phi_T'.